Question 1

Rectangle with breadth as a percent of length: The breadth of a rectangle equals 60% of its length. If the perimeter is 800 m, find the area (in sq. m) of the field.

Accepted Answer

Introduction / Context:Linking perimeter to dimensions is standard: perimeter fixes the sum of length and breadth. Given a percentage relation between breadth and length, we can solve for both and compute area. Given Data / Assumptions: B = 0.60L. Perimeter P = 800 m ⇒ 2(L + B) = 800. Area A = L * B. Concept / Approach:Use the perimeter equation to find L + B, substitute B = 0.60L, solve for L, then compute B and multiply to obtain area. Step-by-Step Solution: 2(L + B) = 800 ⇒ L + B = 400.With B = 0.60L: L + 0.60L = 400 ⇒ 1.60L = 400 ⇒ L = 250 m.Then B = 0.60 * 250 = 150 m.Area A = L * B = 250 * 150 = 37500 sq. m. Verification / Alternative check: Perimeter check: 2(250 + 150) = 2 * 400 = 800 ✔ Why Other Options Are Wrong: 47500, 57500, 77500 come from incorrect algebra or misusing percentages. Common Pitfalls: Using 60 as a fraction instead of 0.60.Forgetting to halve the perimeter to get L + B. Final Answer: 37500

Question 2

Usable typing area with margins: A sheet is 20 cm × 30 cm (used lengthwise). Margins: 2 cm on each left/right side and 3 cm on top/bottom. What percentage of the page area is usable for typing?

Accepted Answer

Introduction / Context:Page layout questions ask you to subtract margins from the total dimensions to find the printable/typing area, then express it as a percent of the full page area. Given Data / Assumptions: Total page: 20 cm × 30 cm. Margins: 2 cm each side (left and right), 3 cm top and bottom. “Lengthwise” orientation does not change the arithmetic; it simply clarifies which side is treated as vertical. Concept / Approach:Usable width = total width − (left + right). Usable height = total height − (top + bottom). Usable area divided by total area × 100 gives the desired percentage. Step-by-Step Solution: Total area = 20 × 30 = 600 cm^2.Usable width = 20 − 2 − 2 = 16 cm.Usable height = 30 − 3 − 3 = 24 cm.Usable area = 16 × 24 = 384 cm^2.Percentage used = (384 / 600) × 100% = 64%. Verification / Alternative check: Quick ratio check: 16/20 = 0.8 and 24/30 = 0.8 ⇒ area factor 0.8 * 0.8 = 0.64 = 64%. Why Other Options Are Wrong: 74%, 84%, and 94% would require different margin sizes or arithmetic; none match the correct 0.64 factor. Common Pitfalls: Subtracting margins only once instead of both sides (left and right; top and bottom). Final Answer: 64%

Question 3

Circle relation: circumference minus semidiameter equals 37 cm. If (circumference − semidiameter) = 37 cm, find the diameter of the circle (in cm).

Accepted Answer

Introduction / Context:This mixes a linear expression of circle measures. The “semidiameter” is the radius. We are given 2πr − r and asked to find the diameter (2r). Given Data / Assumptions: Let radius = r (cm). Circumference C = 2πr. Given: C − r = 37. Concept / Approach:Form the equation r(2π − 1) = 37 and solve for r. Use the standard approximation π = 22/7 (which makes the arithmetic exact here), then compute the diameter 2r. Step-by-Step Solution: 2πr − r = 37 ⇒ r(2π − 1) = 37.Using π = 22/7 ⇒ 2π − 1 = 44/7 − 1 = 37/7.Then r * (37/7) = 37 ⇒ r = 37 * 7 / 37 = 7 cm.Diameter = 2r = 14 cm. Verification / Alternative check: Check: C = 2πr = 2 * 22/7 * 7 = 44; C − r = 44 − 7 = 37 ✔ Why Other Options Are Wrong: 18, 12, 16 cm do not satisfy the exact difference requirement when substituted. Common Pitfalls: Confusing “semidiameter” with “semicircumference”. It means radius. Final Answer: 14 cm

Question 4

Wheel radius from travel distance and revolutions: A wheel makes 1000 revolutions while covering 88 km without slipping. Find the wheel’s radius (in metres).

Accepted Answer

Introduction / Context:When a wheel rolls without slipping, the linear distance traveled equals the number of revolutions times the circumference. From total distance and revolutions, we can find the circumference and then the radius. Given Data / Assumptions: Total distance D = 88 km = 88,000 m. Number of revolutions n = 1000. Circumference C = 2πr. Concept / Approach:Compute the circumference per revolution as D / n, then solve for r from 2πr = C. Step-by-Step Solution: C = D / n = 88,000 m / 1000 = 88 m per revolution.2πr = 88 ⇒ r = 88 / (2π) = 44 / π.Using π ≈ 22/7 ⇒ r = 44 / (22/7) = 44 * 7 / 22 = 14 m. Verification / Alternative check: Circumference 2π * 14 ≈ 88 m; 1000 such revolutions cover 88,000 m = 88 km. Why Other Options Are Wrong: 13, 12, 11 m produce circumferences not matching 88 m, hence total distance differs from 88 km. Common Pitfalls: Forgetting to convert km to m; using diameter instead of radius in the 2πr formula. Final Answer: 14

Question 5

Triangle area via Heron’s formula: Find the area (in sq. cm) of a triangle with side lengths 26 cm, 24 cm, and 10 cm.

Accepted Answer

Introduction / Context:When three sides are known, Heron’s formula is the standard method to compute area without explicitly finding heights or angles. Given Data / Assumptions: Sides: a = 26 cm, b = 24 cm, c = 10 cm. All satisfy triangle inequality (sum of any two exceeds the third). Concept / Approach:Heron’s formula: Area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c)/2 is the semiperimeter. Step-by-Step Solution: s = (26 + 24 + 10)/2 = 60/2 = 30.Area^2 = 30 * (30 − 26) * (30 − 24) * (30 − 10) = 30 * 4 * 6 * 20.Compute: 30 * 4 = 120; 6 * 20 = 120; 120 * 120 = 14,400.Area = √14,400 = 120 sq. cm. Verification / Alternative check: A 10–24–26 is not a right triangle (since 10^2 + 24^2 = 100 + 576 = 676 = 26^2), actually it is right-angled! Thus area = (1/2)*10*24 = 120, confirming Heron’s result. Why Other Options Are Wrong: 108, 112, 116 do not match √14,400 and also contradict the right-triangle shortcut result. Common Pitfalls: Arithmetic slips in multiplying the four terms or taking the square root. Final Answer: 120 sq.cm

Question 6

Square from a rectangle’s diagonal: A rectangle has length 3 cm and breadth 4 cm. A square has side equal to the rectangle’s diagonal. Find the area of this square (in sq. cm).

Accepted Answer

Introduction / Context:Rectangles with sides 3 and 4 form a classic 3–4–5 right triangle for the diagonal. The square built on that diagonal has side equal to the diagonal’s length. Given Data / Assumptions: Rectangle: L = 3 cm, B = 4 cm. Square side s = diagonal of rectangle. Concept / Approach:Diagonal d = √(L^2 + B^2). Area of square = s^2 = d^2. Step-by-Step Solution: d = √(3^2 + 4^2) = √(9 + 16) = √25 = 5 cm.Square area = s^2 = 5^2 = 25 sq. cm. Verification / Alternative check: The Pythagorean triple (3, 4, 5) ensures no rounding. Square area equals diagonal squared, which is 25. Why Other Options Are Wrong: 16, 9, 4 correspond to squaring 4, 3, or 2 — not the diagonal length. Common Pitfalls: Mistaking the square’s side as one of the rectangle’s sides rather than the diagonal. Final Answer: 25 sq.cm

Question 7

Fencing three sides of a rectangle: A rectangular field has one 20 ft side left unfenced. If the area is 680 sq. ft, how many feet of fencing are required for the other three sides?

Accepted Answer

Introduction / Context:We know the area and that one side of 20 ft is open (no fence). Using area, we can find the other dimension; then the fencing equals the sum of the remaining three sides. Given Data / Assumptions: One side (length) = 20 ft (unfenced). Area = 680 sq. ft ⇒ 20 × width = 680. Fencing needed = the other 20 ft side + both widths. Concept / Approach:Compute width from area, then sum three sides (20 + width + width). Avoid mixing up which side is open; the open side is specifically the 20 ft side. Step-by-Step Solution: width w = 680 / 20 = 34 ft.Fencing required = 20 + 34 + 34 = 88 ft. Verification / Alternative check: Full perimeter would be 2(20 + 34) = 108, minus the open 20 gives 88; matches. Why Other Options Are Wrong: 22, 44, 66 are partial sums; they do not account for all three fenced sides. Common Pitfalls: Subtracting the wrong side from the full perimeter or miscalculating width from area. Final Answer: 88

Question 8

Rectangle with breadth 75% of length: The breadth is 75% of the length. If the rectangle’s perimeter is 1050 m, find its area (in sq. m).

Accepted Answer

Introduction / Context:Similar to related problems, a perimeter and a fixed proportional relation between sides allow us to solve for each dimension and then compute area. Given Data / Assumptions: B = 0.75L. Perimeter P = 1050 ⇒ 2(L + B) = 1050 ⇒ L + B = 525. Concept / Approach:Substitute B = 0.75L into L + B = 525 to solve for L, then compute B and multiply to get the area. Step-by-Step Solution: L + 0.75L = 525 ⇒ 1.75L = 525 ⇒ L = 300 m.B = 0.75 × 300 = 225 m.Area = L * B = 300 * 225 = 67,500 sq. m. Verification / Alternative check: Perimeter check: 2(300 + 225) = 2(525) = 1050 ✔ Why Other Options Are Wrong: 65700, 54500, 78700 are not the product of 300 and 225 and typically stem from arithmetic mistakes. Common Pitfalls: Treating 75% as 0.75 is essential; mixing percentage handling often causes errors. Final Answer: 67500 sq.m

Question 9

Change in total area of four walls: Room dimensions are in ratio L:B:H = 7:3:1. If breadth and height are doubled while length is halved, by what percent does the total area of the four walls change?

Accepted Answer

Introduction / Context:The total area of four walls (lateral surface area) of a rectangular room equals perimeter of base times height: 2(L + B) * H. We are to compare original and modified wall areas using given dimension changes. Given Data / Assumptions: Original L:B:H = 7k : 3k : 1k. Modified: L' = (1/2)*7k = 3.5k, B' = 2*3k = 6k, H' = 2*1k = 2k. Concept / Approach:Compute original area A1 = 2(L + B)H and new area A2 = 2(L' + B')H'. Then find percentage increase = ((A2 − A1)/A1)*100. Step-by-Step Solution: A1 = 2(7k + 3k)*1k = 2(10k)*k = 20k^2.A2 = 2(3.5k + 6k)*2k = 2(9.5k)*2k = 38k^2.Increase = 38k^2 − 20k^2 = 18k^2.Percent increase = (18 / 20)*100% = 90%. Verification / Alternative check: Use k = 1 to numerically verify: Original 20, New 38 ⇒ increase 18 ⇒ 90%. Why Other Options Are Wrong: 88%, 85%, 84% result from miscomputing either the halving or doubling steps in the formula. Common Pitfalls: Using surface area of all six faces instead of only the four walls.Forgetting the factor 2(L + B) rather than L + B for the perimeter. Final Answer: 90%

Question 10

Units in a 13 by 9 rectangle: How many square units are contained in a rectangle of side lengths 13 units and 9 units?

Accepted Answer

Introduction / Context:The area of a rectangle equals length times breadth. With integer sides, the product gives the number of square unit tiles that fit exactly. Given Data / Assumptions: Length = 13 units. Breadth = 9 units. Concept / Approach:Area A = L * B. Multiply directly since both are in the same unit. Step-by-Step Solution: A = 13 * 9 = 117 square units. Verification / Alternative check: A quick grid visualization: 13 columns × 9 rows has 117 cells; consistent. Why Other Options Are Wrong: 13 and 9 are side lengths, not areas; 13/9 is irrelevant to area computation. Common Pitfalls: Confusing perimeter with area; perimeter would be 2(13 + 9) = 44, not listed here. Final Answer: 117

Question 11

In two triangles, the ratio of their areas is 4 : 3. The ratio of their corresponding heights is 3 : 4. What is the ratio of their corresponding bases?

Accepted Answer

Introduction / Context:Ratios of triangle areas, heights, and bases are tightly linked by the standard area formula. When two triangles are compared using corresponding elements, proportional reasoning quickly yields the missing ratio without requiring actual numeric side lengths. Given Data / Assumptions: Area ratio: A1 : A2 = 4 : 3. Height ratio (corresponding heights): h1 : h2 = 3 : 4. Bases compared are the bases corresponding to those heights. Concept / Approach:For any triangle, area A = (1/2) * base * height. For two triangles using corresponding base and height, A1/A2 = (b1 * h1) / (b2 * h2). Therefore b1/b2 = (A1/A2) / (h1/h2). Step-by-Step Solution: A1/A2 = 4/3h1/h2 = 3/4b1/b2 = (A1/A2) / (h1/h2)b1/b2 = (4/3) / (3/4) = (4/3) * (4/3) = 16/9 Verification / Alternative check:Pick convenient numbers: let h1 = 3, h2 = 4. Choose bases b1 and b2 so that areas match the 4:3 ratio. If b1/b2 = 16/9, then A1 : A2 = (1/2)*b1*3 : (1/2)*b2*4 = (3b1) : (4b2) = 3*(16k) : 4*(9k) = 48k : 36k = 4 : 3, confirming consistency. Why Other Options Are Wrong: 13:9: Does not satisfy (A1/A2)/(h1/h2) relationship. 15:9: Simplifies to 5:3, still inconsistent with required 16:9. 14:9: No proportional derivation yields this pair. Common Pitfalls: Multiplying the given ratios instead of dividing, which would produce 12:12 erroneously. Inverting the height ratio, leading to 9:16. Final Answer:16:9

Question 12

The area of a circular field is 13.86 hectares (1 hectare = 10,000 m^2). Find the total fencing cost at Rs 4.40 per metre (use circumference at the boundary).

Accepted Answer

Introduction / Context:This problem connects area of a circle with its perimeter (circumference) to determine the cost of fencing a circular field. Converting hectares to square metres is essential before using circle formulas. Given Data / Assumptions: Area A = 13.86 hectares = 13.86 * 10,000 = 138,600 m^2. Cost per metre = Rs 4.40. Fencing length required = circumference = 2 * pi * r. Concept / Approach:Use A = pi * r^2 to find r. Then circumference C = 2 * pi * r. A direct composite relation eliminates r: C = 2 * sqrt(A * pi). Multiply by cost per metre to get total cost. Step-by-Step Solution: A = 138,600 m^2C = 2 * sqrt(A * pi)Using pi ≈ 22/7 gives C = 2 * sqrt(138,600 * 22/7) = 2 * sqrt(138,600 * 3.142857...) ≈ 2 * 660 = 1,320 mTotal cost = 1,320 * 4.40 = Rs 5,808 Verification / Alternative check:Compute radius first: r = sqrt(A / pi) ≈ sqrt(138,600 / 3.142857...) ≈ 210 m. Then C = 2 * pi * r ≈ 2 * 3.142857... * 210 = 1,320 m. Cost again: 1,320 * 4.40 = Rs 5,808, consistent. Why Other Options Are Wrong: 2808, 3808, 4808: Each corresponds to a much shorter circumference than a 138,600 m^2 circle would have, yielding a radius far too small. Common Pitfalls: Forgetting the hectare to m^2 conversion (factor 10,000). Using diameter instead of radius in circumference formula. Premature rounding that shifts the final cost band. Final Answer:5808

Question 13

A semicircular window has diameter 63 cm. Find its perimeter (the curved edge plus the straight diameter).

Accepted Answer

Introduction / Context:Perimeter of a semicircle consists of the semicircular arc plus the straight diameter edge. Recognizing this avoids the common mistake of using only the curved length. Given Data / Assumptions: Diameter D = 63 cm. Radius r = D/2 = 31.5 cm. Perimeter P = arc length (half circumference) + diameter. Concept / Approach:Full circumference is 2 * pi * r. Half of that is pi * r. Therefore P = pi * r + D. Using pi = 22/7 works neatly with multiples of 7. Step-by-Step Solution: r = 31.5 cm = 63/2 cmArc length = pi * r = (22/7) * (63/2) = (22 * 9) / 2 = 99 cmPerimeter P = arc + diameter = 99 + 63 = 162 cm Verification / Alternative check:Compute half circumference as (pi * D)/2 + D = (22/7 * 63)/2 + 63 = (198)/2 + 63 = 99 + 63 = 162 cm. Same result confirms consistency. Why Other Options Are Wrong: 132, 142, 172: Each misses the exact arithmetic from using pi = 22/7 with D divisible by 7. 132 would be the half circumference only if r were 42, which it is not. Common Pitfalls: Forgetting to add the diameter after computing the arc. Using 2 * pi * r instead of pi * r for the arc portion. Final Answer:162

Question 14

A triangle and a parallelogram are constructed on the same base and have equal areas. If the altitude of the parallelogram is 100 m, find the altitude of the triangle.

Accepted Answer

Introduction / Context:When a triangle and a parallelogram share the same base, comparing their areas reduces to comparing altitudes. This classic relation helps convert one altitude to the other directly. Given Data / Assumptions: Same base length b for both shapes. Parallelogram altitude h_p = 100 m. Areas are equal. Concept / Approach:Area of triangle: A_t = (1/2) * b * h_t. Area of parallelogram: A_p = b * h_p. With A_t = A_p and same base b, the altitudes relate by (1/2) * b * h_t = b * h_p, hence h_t = 2 * h_p. Step-by-Step Solution: A_t = (1/2) * b * h_tA_p = b * h_pEquate: (1/2) * b * h_t = b * 100Cancel b: (1/2) * h_t = 100Therefore h_t = 200 m Verification / Alternative check:Pick b = 10 m. Then A_p = 10 * 100 = 1,000 m^2. For triangle to have 1,000 m^2: (1/2) * 10 * h_t = 1,000 ⇒ h_t = 200 m. Confirms result. Why Other Options Are Wrong: 300m, 400m: Would make the triangle area larger than the parallelogram. 100m: Would make the triangle area half of the parallelogram area, not equal. Common Pitfalls: Forgetting that the triangle area has the 1/2 factor while the parallelogram does not. Assuming altitudes are equal when areas are equal. Final Answer:200m

Question 15

In an obtuse triangle, two sides have lengths 8 and 12, and the included angle is 150 degrees. Find the area of the triangle.

Accepted Answer

Introduction / Context:For a triangle with two known sides and an included angle, the area follows a direct formula using the sine of the included angle. This is especially convenient for non-right triangles such as obtuse triangles. Given Data / Assumptions: Side a = 8, side b = 12. Included angle C = 150 degrees. Units are abstract area units; final answer should be in square units. Concept / Approach:Area of triangle with two sides and included angle: A = (1/2) * a * b * sin(C). Use the exact sine value for 150 degrees: sin(150) = sin(30) = 1/2. Step-by-Step Solution: A = (1/2) * 8 * 12 * sin(150)sin(150) = 1/2A = (1/2) * 96 * (1/2) = 48 * (1/2) = 24 Verification / Alternative check:As a reasonableness check, the product (1/2)*a*b is 48, and multiplying by a sine between 0 and 1 gives an area below 48. Since sin(150) = 0.5, area equals 24, consistent with expectations for an obtuse case with a large included angle but moderate sides. Why Other Options Are Wrong: 48 sq units: Would correspond to sin(150) = 1, not possible. 12 sq units or 6 sq units: Would require a smaller sine value than 0.5. Common Pitfalls: Using degrees in a calculator set to radians resulting in incorrect sine values. Confusing 150 degrees with 30 degrees in the triangle but forgetting that the sine is the same; still the included angle is 150, so the formula remains valid with sin(150) = 0.5. Final Answer:24 sq units

Question 16

A 120-degree sector cut from a circle has area 66/7 square centimetres. Find the radius of the circle.

Accepted Answer

Introduction / Context:Sector area links a portion of a circle to its radius through the central angle fraction. By equating the given sector area to the fractional part of the full circle area, the radius follows directly. Given Data / Assumptions: Sector angle = 120 degrees. Sector area = 66/7 cm^2. Full circle area = pi * r^2. Concept / Approach:The sector with central angle theta has area (theta/360) * pi * r^2. With theta = 120, this factor is 120/360 = 1/3. So (1/3) * pi * r^2 = 66/7. Solve for r using a convenient rational value for pi = 22/7 to keep arithmetic clean. Step-by-Step Solution: (1/3) * pi * r^2 = 66/7pi * r^2 = 3 * (66/7) = 198/7Using pi = 22/7 gives (22/7) * r^2 = 198/7r^2 = 198 / 22 = 9r = 3 cm Verification / Alternative check:Back-substitute: Full area = pi * r^2 ≈ (22/7) * 9 = 198/7. Sector is one third of that: (1/3) * 198/7 = 66/7 cm^2, matching the given value exactly. Why Other Options Are Wrong: 1cm or 2cm: Lead to sector areas far below 66/7. 4cm: Produces a sector area larger than 66/7 with the given angle. Common Pitfalls: Using 120/180 instead of 120/360 for the sector fraction. Mixing centimetres and metres; units must remain consistent. Final Answer:3cm

Question 17

Floor paving cost for a rectangular room The length of a room is 5.5 m and the width is 3.75 m. Calculate the total cost of paving the entire floor with slabs at the rate of ₹800 per square metre. Give the final amount in rupees (₹).

Accepted Answer

Introduction / Context:Questions on floor paving or tiling costs combine basic mensuration with a straightforward cost calculation. We first find the area of the surface and then multiply by the rate per unit area to obtain the total cost. Given Data / Assumptions: Length (L) = 5.5 m Width (B) = 3.75 m Rate = ₹800 per square metre Entire floor is to be paved; no wastage considered unless stated. Concept / Approach:The cost = Area * Rate. For a rectangle, Area = L * B. Units must be consistent in square metres since the rate is per m^2. Step-by-Step Solution: Area = 5.5 * 3.75 = 20.625 m^2 Cost = 20.625 * 800 = 16,500 Verification / Alternative check:Since 5.5 * 4 ≈ 22 and width is slightly less than 4 (3.75), the area should be slightly below 22, i.e., 20.625. Multiplying by 800 yields ₹16,500, which aligns with the calculation. Why Other Options Are Wrong:₹15,500, ₹17,500, and ₹18,500 do not match the exact area * rate computation. “None of these” is unnecessary since a matching option exists. Common Pitfalls:Mixing units or rounding too early (e.g., approximating 3.75 as 3.8) can lead to small yet decisive errors in cost. Always compute the exact area first. Final Answer:₹16,500

Question 18

Longest pole inside a cuboidal room What is the maximum possible length of a pole that can be placed inside a room of dimensions 12 m (length), 4 m (breadth), and 3 m (height)?

Accepted Answer

Introduction / Context:The longest rod/pole that fits inside a cuboidal room equals the space diagonal of the cuboid. This is a classic 3D mensuration result derived from the Pythagorean theorem applied twice. Given Data / Assumptions: Length = 12 m, Breadth = 4 m, Height = 3 m Pole length sought = space diagonal of cuboid Concept / Approach:Space diagonal d of a cuboid with dimensions l, b, h is given by d = sqrt(l^2 + b^2 + h^2). Step-by-Step Solution: d = sqrt(12^2 + 4^2 + 3^2) = sqrt(144 + 16 + 9) = sqrt(169) = 13 m Verification / Alternative check:The diagonal of the floor is sqrt(12^2 + 4^2) = sqrt(160). Extending to 3D with height 3 gives sqrt(160 + 9) = sqrt(169) = 13, confirming the result. Why Other Options Are Wrong:14 m, 15 m, and 16 m exceed the exact space diagonal and cannot fit entirely within the room. Common Pitfalls:Confusing the space diagonal with a face diagonal (e.g., sqrt(12^2 + 4^2)) leads to an underestimate. Always include all three dimensions. Final Answer:13 m

Question 19

Squares with diagonals in the ratio 2:5 The diagonals of two squares are in the ratio 2:5. Find the ratio of their areas.

Accepted Answer

Introduction / Context:Area comparisons for similar figures depend on the square of a linear scale factor. For squares, side and diagonal are linearly related, so area is proportional to the square of the diagonal as well. Given Data / Assumptions: Diagonal ratio = 2 : 5 Two perfect squares Concept / Approach:If a linear dimension scales by k, the area scales by k^2. Hence, Area ratio = (Diagonal ratio)^2. Step-by-Step Solution: Area ratio = (2/5)^2 = 4/25 ⇒ 4 : 25 Verification / Alternative check:Let diagonals be 2d and 5d. Sides are (2d)/√2 and (5d)/√2. Their area ratio is ( (2d)^2 / 2 ) : ( (5d)^2 / 2 ) = 4d^2/2 : 25d^2/2 = 4 : 25. Why Other Options Are Wrong:2:25, 3:25, and 6:25 are not squares of 2:5; the correct square ratio is 4:25. Common Pitfalls:Using the ratio directly (2:5) instead of squaring it is a frequent error in similarity-based problems. Final Answer:4:25

Question 20

Equilateral triangle described on the diagonal of a square A square has side a. An equilateral triangle is constructed on the square’s diagonal as its side. What is the ratio of the area of the equilateral triangle to the area of the square?

Accepted Answer

Introduction / Context:Here, the side of the equilateral triangle equals the diagonal of the square. Converting side relations into areas gives a clean ratio without needing the actual numeric value of a. Given Data / Assumptions: Square side = a ⇒ diagonal d = a*√2 Equilateral triangle side = d Concept / Approach:Area(square) = a^2. Area(equilateral) = (√3/4)*side^2. Substitute side = a√2. Step-by-Step Solution: Area(tri) = (√3/4) * (a√2)^2 = (√3/4) * 2a^2 = (√3/2) * a^2 Area(square) = a^2 Ratio = (√3/2) : 1 = √3 : 2 Verification / Alternative check:Let a = 2 for ease: diagonal = 2√2; triangle area = (√3/4)*(8) = 2√3; square area = 4 ⇒ ratio 2√3 : 4 = √3 : 2. Why Other Options Are Wrong:1:2, 1:3, and 2:3 do not arise from the exact formula for the equilateral triangle’s area based on the diagonal length. Common Pitfalls:Confusing the diagonal with the side of the square or forgetting to square the √2 factor when computing area of the triangle can derail the answer. Final Answer:√3:2

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