Question 1

An angle is equal to two-thirds of its supplementary angle. What is the measure of this angle in degrees?

Accepted Answer

Introduction / Context: This is a simple algebraic problem about supplementary angles. Two angles are supplementary if their sum is 180 degrees. The question tells you that one angle is two-thirds of its supplementary angle, and you must find the measure of that angle. Such problems help reinforce the relationship between angles that form a straight line and your ability to set up equations from word statements. Given Data / Assumptions: Let the required angle be x degrees. Its supplementary angle is 180° - x. The given relationship is: x is two-thirds of its supplementary angle. We must find the value of x in degrees. Concept / Approach: Use the definition of supplementary angles and translate the phrase "two-thirds of its supplementary angle" into an equation. The basic idea is: x = (2 / 3) * (180° - x). Then solve this linear equation in x. This approach is very standard for problems involving fractions of angles and their complements or supplements. Step-by-Step Solution: Step 1: Let the required angle be x degrees. Step 2: Its supplementary angle is 180° - x. Step 3: The condition given is: x = (2 / 3) * (180° - x). Step 4: Multiply both sides by 3 to clear the fraction: 3x = 2 * (180° - x). Step 5: Expand the right side: 3x = 360° - 2x. Step 6: Add 2x to both sides: 5x = 360°. Step 7: Divide both sides by 5: x = 360° / 5 = 72°. Step 8: Therefore, the angle measures 72 degrees. Verification / Alternative check: Compute the supplementary angle: 180° - 72° = 108°. Now check the relationship: two-thirds of 108° is (2 / 3) * 108° = 72°, which matches x. This quick check confirms that our algebraic equation accurately captured the given condition and that the arithmetic steps were performed correctly. Why Other Options Are Wrong: If x = 108°, then its supplementary angle is 72°, and x would be greater than its supplement, not two-thirds of it. If x = 54°, then its supplement is 126°, and (2 / 3) * 126° = 84°, which is not 54°. For x = 36°, the supplement is 144°, and (2 / 3) * 144° = 96°, not 36°. An angle of 90° would have a supplement of 90°, and two-thirds of 90° is 60°, not 90°. Only 72° satisfies the given relationship exactly. Common Pitfalls: One common error is reversing the relationship and writing 180° - x = (2 / 3)x instead of x = (2 / 3)(180° - x), which leads to a different equation and the wrong answer. Another mistake is adding the angles to get 90° instead of 180°, confusing supplementary with complementary angles. Some students also mishandle the fraction when multiplying across the equation, leading to incorrect coefficients. Writing the equation carefully from the English description is the key step. Final Answer: The angle whose measure is two-thirds of its supplementary angle is 72°.

Question 2

The area of a trapezium is 21 square centimetres, the distance between its two parallel sides is 3 cm, and one of the parallel sides is 8.3 cm. What is the length of the other parallel side?

Accepted Answer

Introduction / Context: This problem deals with the area of a trapezium (trapezoid), which has two parallel sides called bases and a distance between them called height. You are given the area, the height (the distance between the parallel sides), and one base, and you must find the other base. Questions of this type are very typical in geometry-based aptitude tests and help reinforce the area formula for trapeziums. Given Data / Assumptions: Area of the trapezium A = 21 square centimetres. Distance between the parallel sides (height) h = 3 cm. One of the parallel sides, say a, is 8.3 cm. The other parallel side, say b, is unknown. We use the standard area formula for a trapezium. Concept / Approach: The area A of a trapezium with parallel sides a and b and height h is given by: A = (1 / 2) * (a + b) * h. We are given A, h, and a, and we must solve for b. Rearranging the formula allows us to isolate a + b and then find b by subtracting a. Step-by-Step Solution: Step 1: Write the area formula: A = (1 / 2) * (a + b) * h. Step 2: Substitute A = 21, h = 3, and a = 8.3: 21 = (1 / 2) * (8.3 + b) * 3. Step 3: Simplify the factor (1 / 2) * 3 = 3 / 2. Step 4: So 21 = (3 / 2) * (8.3 + b). Step 5: Multiply both sides by 2 / 3 to clear the fraction: 8.3 + b = (21 * 2 / 3). Step 6: Compute (21 * 2 / 3) = 14. Step 7: So 8.3 + b = 14, hence b = 14 - 8.3 = 5.7 cm. Step 8: Therefore, the other parallel side has length 5.7 cm. Verification / Alternative check: Check by plugging back into the area formula. Compute (a + b) = 8.3 + 5.7 = 14. Then area A = (1 / 2) * 14 * 3 = 7 * 3 = 21 square centimetres, which matches the given area exactly. This confirms that the calculated length b = 5.7 cm is correct. Why Other Options Are Wrong: If the other side were 12.7 cm, then a + b would be 21.0, giving area (1 / 2) * 21 * 3 = 31.5, which is too large. If b were 7.7 cm, then a + b = 16, giving area (1 / 2) * 16 * 3 = 24, again too large. For 10.7 cm, the sum is 19, giving area 28.5. A value of 9.7 cm also gives an incorrect area. Only b = 5.7 cm restores the area to 21 square centimetres. Common Pitfalls: A common mistake is to forget the factor of 1 / 2 in the area formula or to treat 8.3 as an approximate 8, which produces incorrect arithmetic. Another error is solving for a + b correctly but then mis-subtracting the known side, leading to a wrong length. Careful handling of decimal subtraction (14 - 8.3) is important to avoid minor but crucial numerical errors. Final Answer: The length of the other parallel side of the trapezium is 5.7 cm.

Question 3

Triangles ABC and PQR are similar, and the ratio of corresponding sides is AB : PQ = 2 : 3. AD is the median to side BC in triangle ABC and PS is the median to side QR in triangle PQR. What is the value of (BD/QS)^2?

Accepted Answer

Introduction / Context: This problem combines the concepts of similar triangles and medians. It tests whether you understand that when triangles are similar, not only their sides but also certain segments such as medians and altitudes are in the same ratio. You must then use that information to compute a squared ratio involving half segments of corresponding sides. Given Data / Assumptions: Triangle ABC is similar to triangle PQR. The ratio of corresponding sides is AB : PQ = 2 : 3. AD is a median to side BC in triangle ABC, so D is the midpoint of BC. PS is a median to side QR in triangle PQR, so S is the midpoint of QR. We must find the value of (BD / QS)^2. Triangles are assumed to be in a Euclidean plane and similarity is strict. Concept / Approach: When two triangles are similar, the ratio of any pair of corresponding linear measures is constant. This applies to sides, medians, and altitudes. Because AD and PS are medians, their lengths are proportional to the corresponding sides: AD / PS = AB / PQ. Also, a median divides the opposite side into two equal segments, so BD is half of BC and QS is half of QR. Since BC and QR are corresponding sides of similar triangles, BC / QR = AB / PQ. Therefore BD / QS has the same ratio as BC / QR. Step-by-Step Solution: Step 1: From similarity, AB / PQ = 2 / 3. Step 2: For similar triangles, the ratio of corresponding sides is constant, so BC / QR = 2 / 3 as well. Step 3: Because AD is a median, BD = BC / 2. Step 4: Because PS is a median, QS = QR / 2. Step 5: Compute BD / QS: BD / QS = (BC / 2) / (QR / 2) = BC / QR. Step 6: So BD / QS = BC / QR = 2 / 3. Step 7: Now compute (BD / QS)^2 = (2 / 3)^2 = 4 / 9. Step 8: Thus, the required value is 4 / 9. Verification / Alternative check: We can verify by thinking numerically. Suppose AB = 2 units and PQ = 3 units. Then take BC = 2k and QR = 3k for some positive k. Then BD = BC / 2 = k and QS = QR / 2 = 1.5k. Therefore BD / QS = k / (1.5k) = 2 / 3. Squaring gives (BD / QS)^2 = 4 / 9, which matches the analytic reasoning and is independent of the particular value of k. Why Other Options Are Wrong: The value 3 / 5 does not arise from the square of the 2 / 3 ratio. The option 2 / 3 is the ratio BD / QS, not its square. The fraction 4 / 7 is unrelated to the given side ratio 2 : 3. The option 5 / 9 also does not correspond to squaring 2 / 3 or any direct consequence of the given similarity ratio. Only 4 / 9 correctly matches (2 / 3)^2. Common Pitfalls: A frequent error is to confuse BD / QS with AB / PQ itself and forget to square the ratio at the end. Some students also mix up which segments are halves of which sides and mistakenly think BD and QS are in a different ratio. Others attempt to introduce unnecessary variables for the medians, which complicates the problem. Remembering that medians to corresponding sides scale in the same way as the sides themselves and that halving both sides cancels out in the ratio is crucial. Final Answer: The value of (BD / QS)^2 is 4/9.

Question 4

AB and AC are two tangents drawn from an external point A to a circle of radius 6 cm. If ∠BAC = 60°, what is the value, in centimetres, of √(AB² + AC²)?

Accepted Answer

Introduction / Context: This is a geometry problem involving tangents drawn from an external point to a circle. The tangents have equal length and form a given angle at the external point. You are asked to find the square root of the sum of the squares of their lengths, which essentially uses properties of isosceles triangles and the Pythagoras theorem, together with a key fact about tangent segments from a common external point. Given Data / Assumptions: A circle has centre O and radius r = 6 cm. AB and AC are tangents drawn from an external point A to the circle. ∠BAC = 60°. We must find √(AB² + AC²). Since AB and AC are tangents from the same point, AB = AC. Concept / Approach: First, note that AB = AC because tangents from a common external point are equal in length. Thus we need √(AB² + AC²) = √(2AB²) = AB * √2. To find AB, we use the geometry of tangents and chords. The angle between two tangents from an external point is supplementary to the angle subtended by the chord joining the points of contact at the centre. We then use this central angle to find the chord length BC, and finally relate BC to AB in triangle ABC, which is isosceles. Step-by-Step Solution: Step 1: Let B and C be the points of contact on the circle, and O be the centre. Step 2: Tangent segments AB and AC are equal, so AB = AC. Step 3: The angle between tangents at A is ∠BAC = 60°. Step 4: The angle at the centre subtended by chord BC is ∠BOC = 180° - ∠BAC = 180° - 60° = 120°. Step 5: In triangle BOC, OB = OC = radius = 6 cm. It is an isosceles triangle. Step 6: Use the cosine rule to find BC: BC² = OB² + OC² - 2 * OB * OC * cos(∠BOC). Step 7: Substitute values: BC² = 6² + 6² - 2 * 6 * 6 * cos(120°). Step 8: cos(120°) = -1/2, so BC² = 36 + 36 - 72 * (-1/2) = 72 + 36 = 108, hence BC = 6√3. Step 9: Now consider triangle ABC. It is isosceles with AB = AC and included angle ∠BAC = 60° and side BC opposite this angle. Step 10: Apply cosine rule in triangle ABC: BC² = AB² + AC² - 2 * AB * AC * cos(60°). Step 11: Let AB = AC = x, then BC² = x² + x² - 2x² * (1 / 2) = 2x² - x² = x². Step 12: Thus BC = x, so x = BC = 6√3 and therefore AB = AC = 6√3. Step 13: Compute √(AB² + AC²) = √((6√3)² + (6√3)²) = √(108 + 108) = √216 = 6√6. Verification / Alternative check: As a check, we can compute AB² + AC² directly. With AB = AC = 6√3, we have AB² = AC² = 108, so the sum is 216. The square root of 216 can be simplified by factoring: 216 = 36 * 6, so √216 = √36 * √6 = 6√6. This matches our final result. Also, the relationship BC = AB in the isosceles triangle with vertex angle 60° is consistent with the law of cosines, further confirming the internal geometry. Why Other Options Are Wrong: The value 4√6 would correspond to a smaller tangent length and would not match the chord length we found. The values 9√3 and 8√3 do not simplify to √(AB² + AC²) when AB = AC = 6√3. The numeric option 12 is not compatible with the computed sum of squares. Only 6√6 is consistent with a careful application of geometry and the Pythagoras theorem. Common Pitfalls: Students often forget that angle between tangents at an external point is supplementary to the angle at the centre, leading to an incorrect central angle. Others confuse the lengths of the tangent segments and the radius, or misapply the cosine rule. Some also try to compute AB directly using right triangles with the centre without correctly identifying the angles. A clear diagram and consistent stepwise application of known formulas avoid these errors. Final Answer: Therefore, the value of √(AB² + AC²) is 6√6.

Question 5

Two circles have radii 3 cm and 4 cm respectively, and the distance between their centres is 10 cm. What is the ratio of the length of a direct common tangent to the length of a transverse (internal) common tangent?

Accepted Answer

Introduction / Context: This question involves geometric properties of tangents drawn to two circles. You are asked to compare the length of a direct common tangent with the length of a transverse (internal) common tangent. The problem checks whether you know the standard formulas relating the distance between centres and the radii of the circles to the lengths of these tangents. Given Data / Assumptions: Radius of the first circle, r1 = 3 cm. Radius of the second circle, r2 = 4 cm. Distance between the centres of the circles, d = 10 cm. We must find the ratio (length of direct common tangent) : (length of transverse common tangent). Both circles are separate and the tangents considered are the usual external and internal common tangents. Concept / Approach: For two circles with radii r1 and r2 and centre distance d, there are formulas for the lengths of the common tangents: Length of a direct (external) common tangent, L_d = √(d^2 - (r1 - r2)^2). Length of a transverse (internal) common tangent, L_t = √(d^2 - (r1 + r2)^2). We simply substitute the given values into these formulas and then form the ratio L_d : L_t, simplifying if possible. Step-by-Step Solution: Step 1: Identify given values: d = 10, r1 = 3, r2 = 4. Step 2: Compute r1 - r2 = 3 - 4 = -1; the square is (r1 - r2)^2 = (-1)^2 = 1. Step 3: Compute r1 + r2 = 3 + 4 = 7; the square is (r1 + r2)^2 = 7^2 = 49. Step 4: Find L_d: L_d = √(d^2 - (r1 - r2)^2) = √(10^2 - 1) = √(100 - 1) = √99. Step 5: Find L_t: L_t = √(d^2 - (r1 + r2)^2) = √(10^2 - 49) = √(100 - 49) = √51. Step 6: Ratio L_d : L_t = √99 : √51. Step 7: Simplify the ratio by factoring: √99 = √(9 * 11) = 3√11 and √51 = √(3 * 17) = √3 * √17. Step 8: Write both in terms of √3: √99 = √3 * √33, and √51 = √3 * √17. Step 9: Cancel the common factor √3, giving ratio = √33 : √17. Verification / Alternative check: You can also keep the ratio as √99 : √51 and compare it with the options. Noticing that 99 and 51 share a factor 3, divide both under the radical by 3 to get √(99 / 3) : √(51 / 3) = √33 : √17. This directly matches option √33 : √17, confirming that the algebraic simplification is correct without needing to evaluate any decimal approximations. Why Other Options Are Wrong: The option √51 : √68 would require the internal tangent to have length √68, which does not match the given formula and data. The ratio √66 : √51 has the two numbers reversed and does not follow from the calculations. The ratio √28 : √17 does not correspond to any combination of d^2, (r1 - r2)^2 or (r1 + r2)^2 for the given values. The option 1 : 1 would mean the direct and transverse tangents have equal length, which only occurs in special symmetric cases not satisfied here. Only √33 : √17 matches the correct ratio. Common Pitfalls: A common mistake is to swap the formulas and use r1 + r2 where r1 - r2 is needed, or vice versa. Another issue is forgetting to square the differences or sums before subtracting from d^2. Some students also simplify the radicals incorrectly, losing common factors and getting a ratio that does not match any option. Careful substitution and stepwise simplification of the radicals is essential for accuracy. Final Answer: The ratio of the length of the direct common tangent to the length of the transverse common tangent is √33 : √17.

Question 6

Triangle ABC has side lengths AB = 5 cm, AC = √41 cm and BC = 8 cm. AD is drawn perpendicular to BC from vertex A. What is the area, in square centimetres, of triangle ABD?

Accepted Answer

Introduction / Context: This question combines coordinate geometry thinking with classical triangle area formulae. You are given all three side lengths of a triangle and an altitude from one vertex to the opposite side. You must find the area of just one of the smaller triangles formed by this altitude. The problem tests understanding of how altitudes split the base and how area is computed using base and height. Given Data / Assumptions: Triangle ABC has sides AB = 5 cm, AC = √41 cm, and BC = 8 cm. AD is the altitude from A to side BC, so AD is perpendicular to BC. D lies on BC, and we consider triangle ABD. We must find the area of triangle ABD. The triangle is non-degenerate and lies in a Euclidean plane. Concept / Approach: We can place the triangle on a coordinate system to find the height AD easily. Let BC be on the x-axis with B at (0, 0) and C at (8, 0). Then the coordinates of A = (x, y) must satisfy the distance conditions AB = 5 and AC = √41. Solving these gives the height y, which equals the length of AD. Once we know the height, the area of triangle ABD is given by one-half the product of base BD and the height AD. Step-by-Step Solution: Step 1: Place B at (0, 0) and C at (8, 0). Step 2: Let A have coordinates (x, y). Step 3: From AB = 5, we get: x² + y² = 5² = 25. Step 4: From AC = √41, we get: (x - 8)² + y² = 41. Step 5: Subtract the first equation from the second: (x - 8)² - x² = 41 - 25 = 16. Step 6: Expand (x - 8)² = x² - 16x + 64, so x² - 16x + 64 - x² = 16. Step 7: Simplify to get -16x + 64 = 16 ⇒ -16x = -48 ⇒ x = 3. Step 8: Substitute x = 3 into x² + y² = 25: 9 + y² = 25 ⇒ y² = 16 ⇒ y = ±4. Step 9: The altitude length AD is the absolute value of the y-coordinate, so AD = 4 cm. Step 10: D lies on BC at the vertical projection of A, so D has coordinates (3, 0). This means BD = 3 cm and DC = 5 cm. Step 11: Area of triangle ABD is (1 / 2) * base * height = (1 / 2) * BD * AD = (1 / 2) * 3 * 4 = 6 square centimetres. Verification / Alternative check: We can verify by computing the total area of triangle ABC using base BC = 8 cm and height AD = 4 cm. The total area is (1 / 2) * 8 * 4 = 16 square centimetres. The segment BD is 3 cm, so area of ABD is (3 / 8) times the total area, because it uses base 3 instead of 8 with the same height. (3 / 8) * 16 = 6 square centimetres, which matches the value we found earlier. This consistency confirms our answer. Why Other Options Are Wrong: An area of 12 or 20 square centimetres is too large considering that the entire triangle only has area 16 square centimetres. An area of 10 square centimetres would again exceed the total area proportionally when compared with the base segment BD = 3 cm on a base of length 8 cm. The option 8 square centimetres would represent half the total area, which would require BD = 4 cm, not the 3 cm derived from the coordinate solution. Only 6 square centimetres fits all the geometric relationships exactly. Common Pitfalls: Students sometimes incorrectly assume that BD is half of BC or make guesses about its length without using the distance conditions. Another error is to misapply the distance formula or to forget that both equations for AB and AC must be satisfied simultaneously. Some also compute the total area correctly but then divide it into two smaller triangle areas in the wrong ratio. A systematic coordinate approach avoids guesswork and leads to the correct values. Final Answer: The area of triangle ABD is 6 square centimetres.

Question 7

A rectangle has length 8 cm and breadth 6 cm. From each of its four vertices, straight cuts are made so that the remaining figure is a regular octagon (all sides equal). What is the length, in centimetres, of each side of this regular octagon?

Accepted Answer

Introduction / Context: In this geometry question, we start with a rectangle whose length is 8 cm and breadth is 6 cm. Each vertex is cut off in such a way that the remaining figure becomes a regular octagon, meaning all eight sides are equal in length. The main challenge is to express the side length of this octagon in terms of the dimensions of the original rectangle and the amounts cut off at the corners. Given Data / Assumptions: Original rectangle: length = 8 cm, breadth = 6 cm. Four corners are cut to form a regular octagon. Let the amount cut off along the longer side at each corner be x cm. Let the amount cut off along the shorter side at each corner be y cm. The resulting octagon has all sides equal in length. Concept / Approach: The new figure has 8 equal sides. These sides come from: The remaining horizontal portions of the length: 8 - 2x. The remaining vertical portions of the breadth: 6 - 2y. The slant (diagonal) edges formed by cutting off the corners: each diagonal has length √(x^2 + y^2). For a regular octagon, all these three kinds of edges must have the same length. This gives us equations relating x, y and the octagon side length a, and solving these equations yields the required side length. Step-by-Step Solution: Let the side length of the octagon be a. Horizontal side length: a = 8 - 2x. Vertical side length: a = 6 - 2y. Diagonal side length: a = √(x^2 + y^2). From a = 8 - 2x and a = 6 - 2y, we get 8 - 2x = 6 - 2y. So 2 = 2x - 2y → x - y = 1 → x = y + 1. Now a = 6 - 2y. Also a^2 = x^2 + y^2. Substitute x = y + 1: a^2 = (y + 1)^2 + y^2 = 2y^2 + 2y + 1. But a = 6 - 2y, so (6 - 2y)^2 = 2y^2 + 2y + 1. Compute left side: (6 - 2y)^2 = 36 - 24y + 4y^2. Set equal: 36 - 24y + 4y^2 = 2y^2 + 2y + 1. Rearrange: 4y^2 - 2y^2 - 24y - 2y + 36 - 1 = 0 → 2y^2 - 26y + 35 = 0. Divide by 1: 2y^2 - 26y + 35 = 0. Solve quadratic: y = [26 ± √(26^2 - 4 * 2 * 35)] / (2 * 2). Compute discriminant: 26^2 = 676, 4 * 2 * 35 = 280, so √(676 - 280) = √396 = √(36 * 11) = 6√11. Thus y = (26 ± 6√11) / 4 = (13 ± 3√11) / 2. We need y to be less than 3 (so that 6 - 2y > 0), so we choose y = (13 - 3√11) / 2. Now a = 6 - 2y = 6 - (13 - 3√11) = 3√11 - 7. Verification / Alternative check: Approximate √11 ≈ 3.316, so 3√11 ≈ 9.948. Then a ≈ 9.948 - 7 = 2.948 cm, which is a reasonable side length (less than both 6 and 8). Also y ≈ (13 - 9.948) / 2 ≈ 1.526 cm, so the trims are small but positive, which is geometrically valid. Why Other Options Are Wrong: Option B (5√13 – 8) gives a much larger side length that would not fit inside the original 8 cm by 6 cm rectangle once corners are cut. Option C (4√7 – 11) and Option D (6√11 – 9) either give negative or unrealistic dimensions when checked numerically. Option E (2√10 – 5) does not satisfy the algebraic relationship derived from equating horizontal, vertical and diagonal sides. Common Pitfalls: A common mistake is to assume the cuts at each corner are squares (x = y), which only works if the original figure is a square, not a rectangle. Another error is to assume that 8 - 2x automatically equals 6 - 2x, which is impossible and leads to contradiction. Students may also forget to check which root of the quadratic is geometrically valid (positive but small enough so that the remaining sides stay positive). Final Answer: Thus, the length of each side of the regular octagon formed from the 8 cm by 6 cm rectangle is 3√11 – 7 centimetres.

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