Question 1

The diagonal of a square is 10 cm long. Using properties of a square, what is the area of this square measured in square centimetres?

Accepted Answer

Introduction / Context: This problem applies basic geometry of squares and right triangles. When the diagonal of a square is known, we can use Pythagoras theorem to relate the diagonal to the side, and then compute the area. Questions like this are common in aptitude tests to check quick reasoning with simple formulas. Given Data / Assumptions: The figure is a perfect square. Length of the diagonal d = 10 cm. All angles in a square are right angles. We are asked to find the area in sq cm. Concept / Approach: For a square of side s, the diagonal d relates to the side by: d = s√2 This comes from Pythagoras theorem on a right triangle formed by two sides and the diagonal. Then area A of the square is: A = s^2 So we first find s from the diagonal, then compute s^2. Step-by-Step Solution: Given d = 10 cm, use d = s√2. So s = d / √2 = 10 / √2. Rationalise: 10 / √2 = (10√2) / 2 = 5√2. Now compute area: A = s^2 = (5√2)^2. (5√2)^2 = 25 * 2 = 50. Therefore, A = 50 sq cm. Verification / Alternative check: We can check quickly by reversing the process. If the side were 5√2 cm, the diagonal would be s√2 = 5√2 * √2 = 5 * 2 = 10 cm, which matches the given diagonal. So our derived side is consistent, and the square of that side is correctly 50. Why Other Options Are Wrong: 100 sq cm and 200 sq cm are too large and would correspond to much bigger side lengths. 25 sq cm would correspond to side 5 cm, which would give a diagonal of 5√2 ≈ 7.07 cm, not 10 cm. Common Pitfalls: Students sometimes mistakenly treat the diagonal as the side and directly square 10 to get 100, ignoring the √2 relationship. Others forget to square the entire expression 5√2, or mis-handle the rationalisation. Always remember that the diagonal of a square is longer than its side by a factor of √2. Final Answer: The area of the square is 50 sq cm.

Question 2

The area of an equilateral triangle is given as 36√3 square centimetres. Using the standard area formula for an equilateral triangle, what is the length of each side of the triangle in centimetres?

Accepted Answer

Introduction / Context: Here we are given the area of an equilateral triangle and asked to determine the side length. This reverses the usual direct application of the area formula and tests your ability to solve the formula for the side when the area is known, a common style in quantitative aptitude exams. Given Data / Assumptions: The triangle is equilateral. Area A = 36√3 sq cm. We must find the side length a in cm. Standard Euclidean geometry and exact surd representation are assumed. Concept / Approach: For an equilateral triangle with side a, its area is: A = (√3 / 4) * a^2 We are given A, so we solve this equation for a. That involves isolating a^2 and taking the positive square root, since side lengths are positive. Step-by-Step Solution: Start with A = (√3 / 4) * a^2. We are told A = 36√3. So 36√3 = (√3 / 4) * a^2. Multiply both sides by 4 to remove the denominator: 4 * 36√3 = √3 * a^2. This gives 144√3 = √3 * a^2. Divide both sides by √3: a^2 = 144. Take square root: a = √144 = 12 cm. Verification / Alternative check: Substitute a = 12 back into the area formula. Then A = (√3 / 4) * 12^2 = (√3 / 4) * 144 = 36√3 sq cm. This matches the given area, so the side length 12 cm is confirmed as correct. Why Other Options Are Wrong: 6 cm would give area (√3 / 4) * 36 = 9√3 sq cm, too small. 18 cm would give (√3 / 4) * 324 = 81√3 sq cm, which is more than double the required area. 24 cm produces an even larger area, far from 36√3. Common Pitfalls: A common error is to forget the factor 1/4 in the formula or to handle the surd √3 incorrectly when solving for a. Some candidates mistakenly take a to be the square root of 36, instead of correctly isolating a^2. Always rearrange the formula step by step to avoid algebraic slips. Final Answer: The side length of the equilateral triangle is 12 cm.

Question 3

A rectangle has a diagonal of length 51 cm and one of its sides is 24 cm. Using the Pythagoras relation, what is the area of this rectangle in square centimetres?

Accepted Answer

Introduction / Context: This problem connects coordinate geometry ideas with mensuration of a rectangle. Knowing the diagonal and one side, we can find the other side using Pythagoras theorem, and then compute the area. It is a common pattern in aptitude tests to indirectly give the second side in this way. Given Data / Assumptions: Shape is a rectangle, so all angles are 90°. Diagonal length d = 51 cm. One side (take it as length) L = 24 cm. Other side (breadth) is unknown, denote it by B. We need the area A = L * B in sq cm. Concept / Approach: In a rectangle, the diagonal, length, and breadth form a right triangle. By Pythagoras theorem: d^2 = L^2 + B^2 We can solve this equation for B, then multiply by L to obtain the area: A = L * B Step-by-Step Solution: Given d = 51 cm and L = 24 cm. Use d^2 = L^2 + B^2. Compute squares: 51^2 = 2601 and 24^2 = 576. So 2601 = 576 + B^2. Therefore, B^2 = 2601 - 576 = 2025. Take square root: B = √2025 = 45 cm. Now area A = L * B = 24 * 45. 24 * 45 = 1080 sq cm. Verification / Alternative check: We can check quickly: 24^2 + 45^2 = 576 + 2025 = 2601, and √2601 = 51, which matches the given diagonal. So the side lengths 24 cm and 45 cm are consistent, and their product gives the correct area. Why Other Options Are Wrong: 540 and 810 are too small and arise if you halve one side or miscompute the breadth. 360 corresponds to 24 × 15, which would give a much shorter diagonal and does not satisfy Pythagoras theorem with 51 cm. Common Pitfalls: Errors often occur when squaring or subtracting large numbers, or when taking the square root. Some students mistakenly use perimeter formulas instead of applying Pythagoras theorem. Always remember that diagonal, length, and breadth form a right triangle in a rectangle. Final Answer: The area of the rectangle is 1080 sq cm.

Question 4

A rectangle has a perimeter of 90 cm and a breadth of 20 cm. Using basic rectangle formulas, what is the area of this rectangle in square centimetres?

Accepted Answer

Introduction / Context: This is a straightforward rectangle mensuration problem. The data is given via the perimeter and one side (breadth), and you are expected to derive the other side (length) and then compute area. Such questions test direct application of perimeter and area formulas for rectangles. Given Data / Assumptions: Perimeter P of the rectangle = 90 cm. Breadth B = 20 cm. Length L is unknown. We are asked to find the area A in sq cm. Concept / Approach: For a rectangle: Perimeter P = 2(L + B) Area A = L * B We can rearrange the perimeter formula to find L, then multiply by B to get the area. Step-by-Step Solution: Given P = 90 and B = 20. Use P = 2(L + B) ⇒ 90 = 2(L + 20). Divide both sides by 2: L + 20 = 45. So L = 45 - 20 = 25 cm. Now compute area A = L * B = 25 * 20. 25 * 20 = 500 sq cm. Verification / Alternative check: We can recompute the perimeter using L = 25 and B = 20. Then P = 2(L + B) = 2(25 + 20) = 2 * 45 = 90 cm. This matches the given perimeter, confirming our values and the resulting area of 500 sq cm. Why Other Options Are Wrong: 400 corresponds to a length of 20 cm (20 * 20), which would give a perimeter of 80 cm, not 90. 250 and 450 correspond to incorrect length values and do not match the given perimeter when checked. Common Pitfalls: A common error is to confuse half the perimeter (L + B) with a single side, or to forget to divide the perimeter by 2. Another mistake is to misread breadth as length, which might lead to incorrect subtraction. Carefully writing down the perimeter and area formulas prevents such errors. Final Answer: The area of the rectangle is 500 sq cm.

Question 5

A circle is inscribed in a square whose diagonal is 6√2 cm. Using properties of a square and its inscribed circle, what is the area of this circle (in square centimetres)?

Accepted Answer

Introduction / Context: This question combines properties of squares with circles inscribed in them (incircles). Knowing the diagonal of the square allows you to find its side length, and hence the diameter and radius of the inscribed circle. Then you can compute the area of the circle using the standard formula πr². Given Data / Assumptions: A square has diagonal length d = 6√2 cm. A circle is inscribed inside the square, touching all four sides. We are asked to find the area of the circle in square centimetres. The square is assumed to be perfect, with right angles and equal sides. Concept / Approach: For a square with side s and diagonal d: d = s√2 ⇒ s = d / √2 For an inscribed circle, the diameter equals the side of the square, so radius r = s / 2. The area of the circle is: Area = πr^2 Step-by-Step Solution: Given d = 6√2. Compute side s = d / √2 = (6√2) / √2 = 6 cm. For the inscribed circle, diameter = s = 6 cm. So radius r = s / 2 = 6 / 2 = 3 cm. Now area of the circle = πr^2 = π * 3^2. That is area = 9 π square centimetres. Verification / Alternative check: We can double check the diagonal: if s = 6 cm, then diagonal should be s√2 = 6√2, which matches the given diagonal. This confirms that side and hence radius are correctly determined. The circle area π * 9 is therefore reliable. Why Other Options Are Wrong: 6 π and 3 π correspond to radii √6 and √3, which do not match the geometry of an incircle in this square. 9√2 π would involve an extra √2 factor and would arise only if the radius were incorrectly taken as 3√2 instead of 3. Common Pitfalls: A typical mistake is to confuse the diagonal with the side, or to treat the diagonal as the diameter of the circle. Another is failing to divide the side by 2 to get the radius. Always remember: inscribed circle diameter equals the side of the square, not the diagonal. Final Answer: The area of the inscribed circle is 9 π square centimetres.

Question 6

Consider the rectangle in the coordinate plane enclosed by the vertical lines x = -4 and x = 2 and the horizontal lines y = -2 and y = 3. What is the area of this rectangle in square units?

Accepted Answer

Introduction / Context: This problem is about finding the area of an axis-aligned rectangle in the coordinate plane. The sides of the rectangle are given as lines of constant x and constant y values. Such questions test your understanding of distance on the number line and how it translates into side lengths for rectangles. Given Data / Assumptions: Vertical boundaries are x = -4 and x = 2. Horizontal boundaries are y = -2 and y = 3. The region enclosed by these lines is a rectangle. We need the area in square units. Concept / Approach: For a rectangle aligned with the axes, its side lengths are simply the differences between the respective x-coordinates and y-coordinates: Width = right x - left x Height = top y - bottom y Then area A is: A = width * height Step-by-Step Solution: Left x-coordinate = -4, right x-coordinate = 2. Width = 2 - (-4) = 2 + 4 = 6 units. Bottom y-coordinate = -2, top y-coordinate = 3. Height = 3 - (-2) = 3 + 2 = 5 units. Now compute area: A = width * height = 6 * 5. Therefore, A = 30 square units. Verification / Alternative check: We can visualise or quickly sketch the rectangle with corners at (-4, -2), (2, -2), (2, 3), and (-4, 3). Counting squares on a grid would show 6 units across and 5 units up, confirming our multiplication and resulting area of 30. Why Other Options Are Wrong: 20 and 25 result from using one correct side (e.g., 5 or 6) but pairing it with an incorrect second side (4 or 5). 40 would come from mistakenly taking one of the side lengths as 8 or misreading coordinates. None of these respect both x and y ranges simultaneously. Common Pitfalls: People sometimes subtract in the wrong order or forget that subtracting a negative adds. Another common oversight is to mix up horizontal and vertical differences. Always compute width from x-values and height from y-values separately. Final Answer: The area of the enclosed rectangle is 30 square units.

Question 7

In triangle ABC, points X and Y are the midpoints of sides AB and AC respectively. If BC + XY = 12 units, then using the mid-segment (midline) theorem, what is the value of BC − XY (in units)?

Accepted Answer

Introduction / Context: This question tests the mid-segment (midline) theorem in triangles. When you join the midpoints of two sides of a triangle, you create a segment that is parallel to the third side and exactly half its length. Using this property, we can relate BC and XY and then solve a simple algebraic equation. Given Data / Assumptions: Triangle ABC is any triangle. X is the midpoint of AB. Y is the midpoint of AC. Segment XY is the line joining these midpoints. BC + XY = 12 units. We must find BC − XY. Concept / Approach: The mid-segment theorem states: XY ∥ BC and XY = (1/2) * BC So if we let BC = b, then XY = b / 2. Given BC + XY in terms of b, we can solve for b, and then compute BC − XY using the same expressions. Step-by-Step Solution: Let BC = b. Then XY = b / 2 (by the mid-segment theorem). We are told BC + XY = 12, so b + (b / 2) = 12. Combine terms: (3b / 2) = 12. Multiply both sides by 2: 3b = 24, so b = 8. Then XY = b / 2 = 8 / 2 = 4. Now BC − XY = 8 − 4 = 4 units. Verification / Alternative check: Check the given condition with b = 8 and XY = 4. Then BC + XY = 8 + 4 = 12, which matches the question, confirming the algebra and the derived value of BC − XY. Why Other Options Are Wrong: 8 units would arise if you misinterpreted XY as equal to BC, not half. 6 units and 2 units come from incorrect algebra or from choosing wrong expressions for XY in terms of BC. Common Pitfalls: Students sometimes forget that the mid-segment is half the corresponding side, or they reverse the fraction and take XY = 2 * BC. Others may incorrectly add or subtract when solving the equation b + b/2 = 12. Carefully write the relation and combine like terms to avoid such mistakes. Final Answer: The value of BC − XY is 4 units.

Question 8

In triangle geometry, the three medians of a triangle intersect at a single special point. This common point of intersection of all three medians is called the triangle's what?

Accepted Answer

Introduction / Context: This is a conceptual geometry question that tests your knowledge of special points associated with a triangle. Each of the four classical centres (centroid, orthocentre, incentre, and circumcentre) is defined in a different way. Here we focus specifically on medians and their point of concurrency. Given Data / Assumptions: We are dealing with a general triangle (no special type assumed). Each median connects a vertex to the midpoint of the opposite side. All three medians intersect in a single point. We must identify the correct name for this point. Concept / Approach: It helps to recall the definitions: Centroid: Intersection point of the three medians. Orthocentre: Intersection point of the three altitudes. Incentre: Intersection point of the three internal angle bisectors. Circumcentre: Intersection point of the perpendicular bisectors of the sides. Recognising which construction is described in the question immediately leads to the correct term. Step-by-Step Solution: The question explicitly mentions medians. By definition, medians connect each vertex to the midpoint of the opposite side. The unique point at which all three medians of a triangle meet is called the centroid. Therefore, the point of intersection of all the three medians is the centroid. Verification / Alternative check: In coordinate geometry, if the vertices of a triangle are (x1, y1), (x2, y2), and (x3, y3), the intersection of the medians is at the point whose coordinates are: ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3) This point is well known as the centroid and lies two-thirds of the way from each vertex along the median, supporting our identification. Why Other Options Are Wrong: Orthocentre involves altitudes, not medians. Incentre uses angle bisectors, and circumcentre uses perpendicular bisectors of the sides. None of these definitions match the description given in the question. Common Pitfalls: Students sometimes mix up the various triangle centres because their names sound similar. The key is to link each centre with its defining construction: medians for centroid, altitudes for orthocentre, and so on. Drawing a simple triangle and sketching the lines helps fix the concept visually. Final Answer: The point of intersection of all the medians is called the centroid of the triangle.

Question 9

In a triangle, the distances from the centroid to the three vertices are 4 cm, 6 cm and 8 cm respectively. Using properties of medians and centroid, what is the length of the smallest median (in centimetres)?

Accepted Answer

Introduction / Context: This triangle problem uses the fact that the centroid divides each median in a fixed ratio. If you know the distance from the centroid to each vertex, you can reconstruct the full median lengths and then identify the shortest one. This is a conceptual geometry question that combines knowledge of triangle centres with simple proportional reasoning. Given Data / Assumptions: G is the centroid of the triangle. Distances from G to the three vertices are 4 cm, 6 cm, and 8 cm. We must find the length of the smallest median of the triangle. All medians are assumed to be drawn correctly from vertices to midpoints of opposite sides. Concept / Approach: A key property of the centroid is: The centroid divides each median in the ratio 2 : 1 from vertex to midpoint. That is, the distance from a vertex to the centroid is 2/3 of the full median length. So for a median of length m, the segment from vertex to centroid is (2/3)m, and from centroid to midpoint is (1/3)m. We can invert this to find m if we know the vertex–centroid distance. Step-by-Step Solution: Let the three medians have lengths m1, m2, and m3. Given vertex–centroid distances are 4 cm, 6 cm, and 8 cm. For each median, vertex–centroid distance = (2/3) * median length. So m1 = (3/2) * 4 = 6 cm. m2 = (3/2) * 6 = 9 cm. m3 = (3/2) * 8 = 12 cm. The smallest median is the one with length 6 cm. Verification / Alternative check: We can verify the ratios quickly: for the median of length 6 cm, 2/3 * 6 = 4 cm, matching one given distance. Similarly, 2/3 * 9 = 6 and 2/3 * 12 = 8. This confirms that we correctly reconstructed all median lengths from the centroid distances. Why Other Options Are Wrong: 8, 7, and 5 do not match any of the median lengths deduced. 8 and 9 might be confused with the given centroid distances or other medians, but the smallest median computed is 6 cm, not 5 cm or 7 cm or 8 cm. Common Pitfalls: It is easy to invert the 2 : 1 ratio incorrectly and use 1/2 or 2 instead of 3/2. Some students also assume the centroid divides the median in half, which is incorrect. Always remember the precise ratio: vertex to centroid is two-thirds of the median, centroid to midpoint is one-third. Final Answer: The smallest median of the triangle has length 6 cm.

Question 10

The two parallel sides of a trapezium are 16 m and 20 m respectively, and the perpendicular distance (height) between them is 10 m. What is the area of this trapezium in square metres?

Accepted Answer

Introduction / Context: This mensuration problem involves a trapezium, a quadrilateral with exactly one pair of parallel sides. The area formula for a trapezium uses the average of the parallel sides multiplied by the height. Such questions check whether you can recall and correctly apply this basic area formula. Given Data / Assumptions: Lengths of the parallel sides are 16 m and 20 m. The height (perpendicular distance between the parallel sides) is 10 m. We are asked for the area in square metres. The shape is a standard trapezium with one pair of parallel sides only. Concept / Approach: The area A of a trapezium with parallel sides a and b, and height h, is: A = (1/2) * (a + b) * h This formula can be seen as taking the average of the two bases and multiplying by the height, analogous to the base–height formula of a rectangle. Step-by-Step Solution: Let a = 16 m, b = 20 m, and h = 10 m. Compute a + b = 16 + 20 = 36. Apply the formula A = (1/2) * (a + b) * h. So A = (1/2) * 36 * 10. (1/2) * 36 = 18, so A = 18 * 10 = 180 square metres. Verification / Alternative check: You can think of the trapezium as a combination of a rectangle and two right triangles. Its average base length is (16 + 20)/2 = 18 m. If you imagine a rectangle of base 18 m and height 10 m, its area is 180 m², matching our trapezium area formula result. Why Other Options Are Wrong: 260, 240, and 360 all arise from misusing the formula, such as forgetting the factor 1/2, adding height incorrectly, or multiplying the bases incorrectly. None of these match the proper computation using the trapezium area formula. Common Pitfalls: A frequent mistake is to multiply just one base by the height, as if the figure were a rectangle, or to take the difference of the bases instead of the sum. Remember always to add the lengths of the parallel sides, divide by 2, and then multiply by the height. Final Answer: The area of the trapezium is 180 m².

Question 11

A rectangle has a perimeter of 48 cm and a breadth of 10 cm. Using the formulae for perimeter and area, what is the area of this rectangle in square centimetres?

Accepted Answer

Introduction / Context: This is another basic rectangle problem, similar in style to many aptitude questions. You are given the perimeter and one side (the breadth) and must determine the area. This tests both your formula knowledge and your algebraic manipulation skills. Given Data / Assumptions: Perimeter P = 48 cm. Breadth B = 10 cm. Length L is unknown. We must find the area A = L * B in sq cm. Concept / Approach: The perimeter P of a rectangle is: P = 2(L + B) From this, we can solve for L: L + B = P / 2 ⇒ L = (P / 2) − B Once we have L, we use: A = L * B Step-by-Step Solution: Given P = 48 and B = 10. Compute P / 2: 48 / 2 = 24. So L + B = 24 ⇒ L = 24 − 10 = 14 cm. Now compute the area: A = L * B = 14 * 10. Thus A = 140 square centimetres. Verification / Alternative check: Check the perimeter with L = 14 and B = 10: P = 2(L + B) = 2(14 + 10) = 2 * 24 = 48 cm, which matches the given perimeter. Therefore the dimensions 14 cm by 10 cm are correct, and the product 140 is indeed the area. Why Other Options Are Wrong: 240 would correspond to a length of 24 cm, giving perimeter 2(24 + 10) = 68 cm, not 48. 480 and 440 are much too large and represent miscalculations like multiplying perimeter by a side or adding values incorrectly. Common Pitfalls: Learners sometimes forget to divide the perimeter by 2 first or inadvertently use a square formula (like side²) as if the rectangle were a square. Writing down the formulas and solving them step by step helps avoid these mistakes. Final Answer: The area of the rectangle is 140 sq cm.

Question 12

The areas of two similar triangles, ΔXYZ and ΔPQR, are 100 sq cm and 25 sq cm respectively. If side PQ = 4 cm in the smaller triangle, what is the corresponding side length XY (in centimetres) in the larger triangle?

Accepted Answer

Introduction / Context: This question is about similar triangles and the relationship between their areas and corresponding side lengths. If two triangles are similar, their areas are proportional to the square of the ratio of their corresponding sides. This concept is widely tested in geometry and aptitude exams. Given Data / Assumptions: Triangles ΔXYZ and ΔPQR are similar. Area of ΔXYZ = 100 sq cm. Area of ΔPQR = 25 sq cm. Side PQ in ΔPQR = 4 cm. PQ corresponds to XY in ΔXYZ. We must find length XY. Concept / Approach: For similar triangles, if k is the ratio of corresponding sides (larger/smaller), then: Area ratio = k^2 Here, area(ΔXYZ) / area(ΔPQR) = 100 / 25 = 4. So k^2 = 4 and k = 2. That means each corresponding side in the larger triangle is twice the corresponding side in the smaller one. Step-by-Step Solution: Given area ratio = 100 / 25 = 4. Let side scale factor (XYZ : PQR) be k. Then k^2 = 4 ⇒ k = 2 (taking positive root, as lengths are positive). Since PQ = 4 cm, and XY corresponds to PQ, we have XY = k * PQ. Thus XY = 2 * 4 = 8 cm. Verification / Alternative check: If all sides scale by factor 2, area scales by factor 2² = 4. A triangle with area 25 sq cm, when scaled by factor 2 in all dimensions, has area 25 * 4 = 100 sq cm, consistent with the given areas. So side 4 cm in the smaller triangle corresponds to a side 8 cm in the larger one. Why Other Options Are Wrong: 16 and 20 cm would correspond to scale factors 4 or 5, which would give area ratios 16 or 25, not 4. 14 cm is also not consistent with any simple square factor relationship between 25 and 100. Common Pitfalls: Many students mistakenly take the area ratio directly as the side ratio, forgetting to take the square root. Always remember: area ratio = (side ratio)², so you must take the square root of the area ratio to find the side ratio. Final Answer: The length of side XY is 8 cm.

Question 13

A regular hexagon (six-sided polygon with all sides and angles equal) has each side of length 6 cm. Using the standard formula, what is the area of this regular hexagon in square centimetres?

Accepted Answer

Introduction / Context: This question involves finding the area of a regular hexagon using its side length. A regular hexagon can be divided into six congruent equilateral triangles, which allows us to use the equilateral triangle area formula. Recognising this decomposition is key to solving such problems quickly. Given Data / Assumptions: The polygon is a regular hexagon. Each side has length 6 cm. We need the total area in sq cm. All internal angles are 120°, and all sides are equal. Concept / Approach: A regular hexagon of side a can be partitioned into 6 equilateral triangles, each with side a. The area of one equilateral triangle with side a is: A_triangle = (√3 / 4) * a^2 Thus, the area of the hexagon is: A_hexagon = 6 * A_triangle = 6 * (√3 / 4) * a^2 Step-by-Step Solution: Let side a = 6 cm. Compute area of one equilateral triangle: A_triangle = (√3 / 4) * a^2. So A_triangle = (√3 / 4) * 6^2 = (√3 / 4) * 36. Simplify: 36 / 4 = 9, so A_triangle = 9√3 sq cm. Area of hexagon = 6 * A_triangle = 6 * 9√3. Therefore, A_hexagon = 54√3 sq cm. Verification / Alternative check: There is also a direct formula for a regular hexagon: A_hexagon = (3√3 / 2) * a^2 Substitute a = 6: A_hexagon = (3√3 / 2) * 36 = 3√3 * 18 = 54√3. This matches the result from the equilateral triangle decomposition, confirming the accuracy. Why Other Options Are Wrong: 27 and 54 with no √3 ignore the surd factor that naturally appears in equilateral triangle area. 27√3 is exactly half the correct result and would come from mistakenly using three triangles instead of six or misapplying the formula. Common Pitfalls: A frequent mistake is to forget that the hexagon splits into six equilateral triangles, not four or three. Another common error is miscomputing the area of a single equilateral triangle, especially the factor (√3 / 4). Careful use of the formula avoids these errors. Final Answer: The area of the regular hexagon is 54√3 sq cm.

Question 14

In coordinate geometry, what is the reflection of the point (2, -3.5) in the y-axis? Give the coordinates of the reflected point.

Accepted Answer

Introduction / Context: This question involves reflections in the coordinate plane, a fundamental concept in analytic geometry. Reflecting a point in the y-axis means producing a mirror image of that point across the vertical axis. Understanding the rule for how coordinates change under such reflections is essential for many geometry and transformation problems. Given Data / Assumptions: Original point is P(2, -3.5). We reflect P across the y-axis. We seek the coordinates of the image point P'. The axes are standard: x-axis horizontal, y-axis vertical. Concept / Approach: Reflection in the y-axis changes the sign of the x-coordinate but leaves the y-coordinate unchanged. In general: If P(x, y) is reflected in the y-axis, then P' = (−x, y). This is because the y-axis is the line x = 0, and reflecting across it flips left and right while keeping vertical position constant. Step-by-Step Solution: We have P(2, -3.5). Under reflection in the y-axis, x → −x and y stays the same. So new x-coordinate = −2. New y-coordinate remains −3.5. Therefore, the reflected point is (−2, −3.5). Verification / Alternative check: Plotting roughly in your mind, point (2, -3.5) lies to the right of the y-axis and below the x-axis. Its reflection should lie at the same vertical level, but the same distance to the left of the y-axis, that is at x = -2. This matches the coordinate (−2, −3.5). Why Other Options Are Wrong: (-2, 3.5) corresponds to reflection in both axes combined (a half-turn), not just the y-axis. (-3.5, -2) and (3.5, -2) swap x and y, which is typical of reflection across the line y = x or similar transformations, not reflection across the y-axis. Common Pitfalls: A common mistake is to flip the wrong coordinate or both coordinates. Another error is to confuse reflection in the x-axis with reflection in the y-axis. Remember: reflection in the y-axis changes the sign of x, reflection in the x-axis changes the sign of y. Final Answer: The reflection of (2, -3.5) in the y-axis is (-2, -3.5).

Question 15

Point A divides the line segment BC internally in the ratio 1 : 3. If the coordinates of B are (4, -4) and the coordinates of C are (0, 6), what are the coordinates of point A?

Accepted Answer

Introduction / Context: This is an internal division problem in coordinate geometry. It uses the section formula, which provides coordinates of a point that divides a line segment joining two given points in a specified ratio. This formula is widely used in analytic geometry and vector problems. Given Data / Assumptions: Point B has coordinates (4, -4). Point C has coordinates (0, 6). Point A divides BC in the ratio 1 : 3. We interpret the ratio as BA : AC = 1 : 3 (internal division). We must find coordinates of A. Concept / Approach: For internal division, if A divides BC in the ratio m : n (BA : AC = m : n), and B(x1, y1), C(x2, y2), then: A(x, y) = ((n x1 + m x2) / (m + n), (n y1 + m y2) / (m + n)) This is the section formula for internal division. Step-by-Step Solution: Here, B(4, -4), C(0, 6), and BA : AC = 1 : 3 ⇒ m = 1, n = 3. Apply the formula for x-coordinate: x = (n x1 + m x2) / (m + n) = (3 * 4 + 1 * 0) / (1 + 3). x = (12 + 0) / 4 = 12 / 4 = 3. Now for y-coordinate: y = (n y1 + m y2) / (m + n) = (3 * (-4) + 1 * 6) / 4. y = (-12 + 6) / 4 = -6 / 4 = -1.5. Therefore A has coordinates (3, -1.5). Verification / Alternative check: We can verify that A is closer to B than to C because the ratio is 1 : 3. B is (4, -4) and C is (0, 6). The x-coordinate 3 is nearer to 4 than to 0, and y-coordinate -1.5 is between -4 and 6, but much closer to -4, consistent with a 1 : 3 division from B to C. Why Other Options Are Wrong: (-3, 1.5) and (-1.5, 3) place A on the opposite side of C or in locations inconsistent with the line segment direction. (1.5, 3) would correspond to a different ratio, not the specified 1 : 3 from B to C. Common Pitfalls: Mixing up m and n, or mistakenly using m with B and n with C in the wrong positions, often leads to incorrect coordinates. Another common issue is misinterpreting the ratio as AC : BA instead of BA : AC. Carefully define which part of the segment each ratio component refers to before applying the formula. Final Answer: The coordinates of point A are (3, -1.5).

Question 16

What is the slope of any line that is parallel to the line passing through the points (4, -2) and (-3, 5) in the coordinate plane?

Accepted Answer

Introduction / Context: This problem uses the concept of slope in coordinate geometry and the fact that all lines parallel to a given non-vertical line share the same slope. The question asks you first to find the slope of a given line and then use this to determine the slope of any line parallel to it. Given Data / Assumptions: One line passes through points A(4, -2) and B(-3, 5). We are looking for the slope of any line parallel to AB. We assume standard Cartesian coordinates. Concept / Approach: The slope m of a line through points (x1, y1) and (x2, y2) is: m = (y2 − y1) / (x2 − x1) If another line is parallel to this one and is not vertical, then its slope is the same number m. Step-by-Step Solution: Take A(4, -2) and B(-3, 5). Compute slope of AB: m = (5 − (-2)) / (-3 − 4). Numerator: 5 − (-2) = 5 + 2 = 7. Denominator: -3 − 4 = -7. So m = 7 / (-7) = -1. Therefore, any line parallel to AB also has slope -1. Verification / Alternative check: We can see that a line with slope -1 rises by 1 unit when moving 1 unit to the left and falls by 1 unit when moving 1 unit to the right. Starting from (4, -2), moving left 7 units goes to (-3, 5) (since y increases by 7). This behaviour is consistent with slope -1 and confirms our calculation. Why Other Options Are Wrong: 3/7 and -3/7 would correspond to lines with much gentler slopes, not matching the steepness of the line from (4, -2) to (-3, 5). A slope of 1 would give a line rising one unit for each move to the right, opposite in sign to the actual line, which slopes downward from left to right. Common Pitfalls: Students sometimes reverse the order of points or miscalculate numerator or denominator signs. Remember that as long as you keep the order consistent (y2 − y1 over x2 − x1), you will get the correct slope. Changing the order of both numerator and denominator simultaneously still yields the same result. Final Answer: The slope of any line parallel to the given line is -1.

Question 17

Triangles ΔXYZ and ΔPQR are similar. The ratio of their perimeters is given as perimeter(ΔXYZ) : perimeter(ΔPQR) = 3 : 2. If side PQ = 6 cm in ΔPQR, then what is the corresponding side length XY (in centimetres) in ΔXYZ?

Accepted Answer

Introduction / Context: This question tests understanding of similar triangles and the relationship between their perimeters and side lengths. For similar triangles, the ratio of their perimeters equals the ratio of any pair of corresponding sides. Given this ratio and one side in the smaller triangle, you can find the corresponding side in the larger triangle. Given Data / Assumptions: ΔXYZ is similar to ΔPQR. Perimeter ratio: perimeter(ΔXYZ) : perimeter(ΔPQR) = 3 : 2. PQ = 6 cm in ΔPQR. Side PQ corresponds to side XY in ΔXYZ. We must find XY. Concept / Approach: For similar triangles, ratio of corresponding sides equals ratio of perimeters. If the perimeter ratio (larger : smaller) is 3 : 2, then each corresponding side in the larger triangle is (3/2) times the corresponding side in the smaller triangle: XY / PQ = 3 / 2 Step-by-Step Solution: Given perimeter(ΔXYZ) : perimeter(ΔPQR) = 3 : 2. Thus side ratio XY : PQ = 3 : 2. We know PQ = 6 cm. So XY / 6 = 3 / 2. Cross multiply: 2 * XY = 3 * 6 = 18. Therefore XY = 18 / 2 = 9 cm. Verification / Alternative check: If every side of ΔPQR is multiplied by factor 3/2 to get ΔXYZ, then the perimeter also multiplies by 3/2, which matches the given perimeter ratio of 3 : 2. This consistency confirms that our side scaling factor and the resulting XY value are correct. Why Other Options Are Wrong: 4 cm would imply a side smaller than PQ, contradicting a larger perimeter for ΔXYZ. 8 cm gives a side ratio of 4/3, which would lead to perimeter ratio 4/3, not 3/2. 12 cm would give a side ratio of 2, implying a perimeter ratio of 2 : 1, again inconsistent with the given 3 : 2. Common Pitfalls: One common error is to invert the ratio and use 2 : 3 instead of 3 : 2, which would give a smaller triangle instead of a larger one. Another pitfall is to confuse perimeter ratio with area ratio; area would scale with the square of the side ratio, not linearly. Final Answer: The length of side XY is 9 cm.

Question 18

In coordinate geometry, what is the reflection of the point (5, 3) across the horizontal line y = -2? Give the coordinates of the reflected point.

Accepted Answer

Introduction / Context: This transformation problem involves reflecting a point across a horizontal line in the coordinate plane. Reflection is a basic geometric transformation where each point is mapped to a mirror image across a given line. Here, the mirror line is y = -2, which is parallel to the x-axis. Given Data / Assumptions: Original point P has coordinates (5, 3). The line of reflection is y = -2. The line y = -2 is horizontal, located 2 units below the x-axis. We must find coordinates of the reflected point P'. Concept / Approach: For reflection across a horizontal line y = k: x-coordinate stays the same. y-coordinate transforms using y' = 2k − y. This formula ensures that the line y = k is exactly halfway between the point and its reflection, so the vertical distances to the line are equal and opposite. Step-by-Step Solution: Given P(5, 3) and reflection line y = k = -2. Keep x the same: x' = 5. Compute new y-coordinate using y' = 2k − y. So y' = 2 * (-2) − 3 = -4 − 3 = -7. Thus, the reflected point P' is (5, -7). Verification / Alternative check: Check the vertical distances. Original y = 3, reflection line y = -2. The distance from P to the line is 3 − (-2) = 5 units. For P' at y = -7, the distance to the line is -2 − (-7) = 5 units. Since these distances are equal and on opposite sides of the line, the reflection is correct. Why Other Options Are Wrong: (-9, 3) and (-9, -3) change the x-coordinate, which should remain 5 under reflection across a horizontal line. (-5, -7) changes both x and y, corresponding to some other transformation, not reflection across y = -2. Common Pitfalls: Learners often confuse reflections across horizontal and vertical lines. For a horizontal line y = k, only the y-coordinate changes; for a vertical line x = k, only the x-coordinate changes. Another mistake is to subtract k from y instead of using the correct 2k − y formula, which ensures the line is the perpendicular bisector of the segment joining point and its image. Final Answer: The reflection of (5, 3) in the line y = -2 is (5, -7).

Question 19

In right triangle LMN, the right angle is at vertex M (∠M = 90°). The measure of angle N is 45°. If the hypotenuse LN has length 9√2 cm, then what is the length (in cm) of side MN?

Accepted Answer

Introduction: This problem involves a right triangle with a 45° angle, which means it is a special 45°–45°–90° triangle. Such triangles have fixed relationships between the lengths of their sides. Recognizing and applying these relationships makes the problem straightforward. Given Data / Assumptions: Triangle LMN is right angled at M, so ∠M = 90°. Angle N is 45°, so angle L is also 45° (sum of angles in a triangle is 180°). The hypotenuse is LN and its length is 9√2 cm. We must find the length of side MN. Concept / Approach: In a 45°–45°–90° triangle, both legs are equal in length. If each leg has length a, then the hypotenuse has length a√2. Hence, given the hypotenuse, we can reverse this relationship: leg length = (hypotenuse) / √2. This eliminates the need for the Pythagorean theorem calculation from scratch. Step-by-Step Solution: Because ∠M = 90° and ∠N = 45°, the third angle ∠L is also 45°. Thus triangle LMN is an isosceles right triangle, with MN = ML. Let each leg be of length a. Then hypotenuse LN = a√2. We are given LN = 9√2 cm, so a√2 = 9√2. Divide both sides by √2 to get a = 9 cm. Thus MN = a = 9 cm. Verification / Alternative check: Check with Pythagoras. If MN = 9 and ML = 9, then LN² = 9² + 9² = 81 + 81 = 162. Hence LN = √162 = √(81 × 2) = 9√2 cm, which matches the given hypotenuse length. The result is correct. Why Other Options Are Wrong: The value 18 cm would make the hypotenuse longer than 9√2 cm. Values 9√2 or 9/√2 do not match the required leg length given the standard 45°–45°–90° ratio. Only 9 cm gives the correct hypotenuse when plugged back into the Pythagorean theorem. Common Pitfalls: A common mistake is to think that the hypotenuse is double a leg rather than a√2 times a leg, or to mix up which side is the hypotenuse. Another frequent error is failing to simplify the expression when dividing by √2. Remember that in a 45°–45°–90° triangle, the hypotenuse is exactly √2 times any leg. Final Answer: The length of MN is 9 cm.

Question 20

In a square, the length of the diagonal is 12 cm. Using properties of squares and right triangles, what is the length (in cm) of each side of the square?

Accepted Answer

Introduction: This question uses the well known relationship between the side and the diagonal of a square. Drawing a diagonal splits the square into two congruent right triangles, and the Pythagorean theorem gives a direct connection between the side length and diagonal length. Given Data / Assumptions: The figure is a square. The diagonal of the square is 12 cm. All sides of a square are equal, and each interior angle is 90°. We must find the side length of the square. Concept / Approach: In a square with side length s, the diagonal d forms the hypotenuse of a right triangle whose legs are both s. By the Pythagorean theorem, d² = s² + s² = 2s², so d = s√2. Solving for s gives s = d / √2. This formula can be used directly once the diagonal is known. Step-by-Step Solution: Let side length of the square be s. Diagonal d = 12 cm. Relationship: d = s√2. So 12 = s√2. Solve for s: s = 12 / √2. Rationalize if desired: 12 / √2 = (12√2) / 2 = 6√2. Therefore, each side of the square is 6√2 cm long. Verification / Alternative check: Check by reversing the relationship: if s = 6√2, then diagonal d should be s√2 = 6√2 * √2 = 6 * 2 = 12 cm, which matches the given diagonal length. Why Other Options Are Wrong: Values 6 cm or 9 cm would give diagonals of 6√2 cm or 9√2 cm, not 12 cm. The option 12√2 cm is far too large for the side, as its diagonal would then be 12√2 * √2 = 24 cm. Only 6√2 cm reproduces the given diagonal of 12 cm. Common Pitfalls: Students sometimes incorrectly assume the diagonal equals 2s instead of s√2, confusing squares with rectangles of special dimensions. Another mistake is forgetting to divide by √2 or rationalizing incorrectly. Always derive the relationship from the Pythagorean theorem if unsure. Final Answer: The length of each side of the square is 6√2 cm.

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