Mensuration Questions and Answers

Q: The area of a square is 4096 sq cm. Find the ratio of the breadth and the length of a rectangle whose length is twice the side of the square and breadth is 24 cm less than the side of the square.

\[ \text{side of square} = \sqrt{4096} = 64 \text{ cm} \]\[ \text{length} = 2 \times 64 = 128 \text{ cm}, \qquad \text{breadth} = 64 - 24 = 40 \text{ cm} \]\[ \text{breadth} : \text{length} = 40 : 128 = 5 : 16 \]That ratio is not among the choices, so the answer is none of these. The 7 : 16 on offer is close enough to catch a careless division.

Q: The diagonal of a square is \(10\sqrt{2}\) cm. What is its area ?

Correct answer: 100 sq. cmFor a square of side \(a\), the diagonal is \(a\sqrt{2}\).\[a\sqrt{2} = 10\sqrt{2} \quad\Longrightarrow\quad a = 10 \text{ cm}\]\[\text{Area} = a^2 = 100 \text{ sq. cm}\]Equivalently, the area of a square is half the square of its diagonal: \(\tfrac{1}{2}(10\sqrt{2})^2 = 100\).

Q: Which of the following has its area and perimeter numerically equal?

Compare each shape's area with its perimeter.Equilateral triangle, side 1: area \( \tfrac{\sqrt{3}}{4} = 0.43 \), perimeter 3Square, side 1: area 1, perimeter 4Regular pentagon, side 1: area 1.72, perimeter 5Square, side 4: area \( 4^{2} = 16 \), perimeter \( 4 \times 4 = 16 \) &check;Only the square of side 4 cm has the two numerically equal. For a square, \( a^{2} = 4a \) forces \( a = 4 \) - a single size, and the reason unit-sided shapes cannot work.

Q: By converting angle from degrees to radians, 105° is equal to

\[ \pi \text{ radians} = 180^{\circ} \]\[ 105^{\circ} = 105 \times \frac{\pi}{180} = \frac{7\pi}{12} \]\[ = 1.832 \text{ rad} \]Since a radian is about \( 57.3^{\circ} \), an angle of \( 105^{\circ} \) should come out a little under 2 radians.

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151.

The area of a square is 4096 sq cm. Find the ratio of the breadth and the length of a rectangle whose length is twice the side of the square and breadth is 24 cm less than the side of the square.

A.18 : 5
B.7 : 16
C.5 : 14
D.None of these.

Answer: D

\[ \text{side of square} = \sqrt{4096} = 64 \text{ cm} \]

\[ \text{length} = 2 \times 64 = 128 \text{ cm}, \qquad \text{breadth} = 64 - 24 = 40 \text{ cm} \]

\[ \text{breadth} : \text{length} = 40 : 128 = 5 : 16 \]

That ratio is not among the choices, so the answer is none of these. The 7 : 16 on offer is close enough to catch a careless division.

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152.

The area of the square formed on the diagonal of a rectangle as its side is 108 1/3 % more than the area of the rectangle. If the perimeter of the rectangle is 28 units, find the difference between the sides of the rectangle?

A.8
B.12
C.6
D.2

Answer: D

Let the sides of the rectangle be l and b respectively.

From the given data,

(√l² + b²) = (1 + 108 1/3 %)lb

=> l² + b² = (1 + 325/3 * 1/100)lb

= (1 + 13/12)lb

= 25/12 lb

=> (l2 + b²)/lb = 25/12

12(l2 + b²) = 25lb

Adding 24lb on both sides

12l2 + 12b² + 24lb = 49lb

12(l2 + b² + 2lb) = 49lb

but 2(l + b) = 28 => l + b = 14

12(l + b)² = 49lb

=> 12(14)² = 49lb

=> lb = 48

Since l + b = 14, l = 8 and b = 6

l - b = 8 - 6 = 2m.

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153.

The diagonal of a square is \(10\sqrt{2}\) cm. What is its area ?

A.50 sq. cm
B.100 sq. cm
C.141 sq. cm
D.200 sq. cm

Answer: B

Correct answer: 100 sq. cm

For a square of side \(a\), the diagonal is \(a\sqrt{2}\).

\[a\sqrt{2} = 10\sqrt{2} \quad\Longrightarrow\quad a = 10 \text{ cm}\]

\[\text{Area} = a^2 = 100 \text{ sq. cm}\]

Equivalently, the area of a square is half the square of its diagonal: \(\tfrac{1}{2}(10\sqrt{2})^2 = 100\).

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154.

Which of the following has its area and perimeter numerically equal?

A.an equilateral triangle of side 1 cm
B.a square of side 1 cm
C.a square of side 4 cm
D.a regular pentagon of side 1 cm.

Answer: C

Compare each shape's area with its perimeter.

Equilateral triangle, side 1: area \( \tfrac{\sqrt{3}}{4} = 0.43 \), perimeter 3
Square, side 1: area 1, perimeter 4
Regular pentagon, side 1: area 1.72, perimeter 5
Square, side 4: area \( 4^{2} = 16 \), perimeter \( 4 \times 4 = 16 \) ✓

Only the square of side 4 cm has the two numerically equal. For a square, \( a^{2} = 4a \) forces \( a = 4 \) - a single size, and the reason unit-sided shapes cannot work.

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155.

By converting angle from degrees to radians, 105° is equal to

A.2.856 rad
B.3.523 rad
C.0.348 rad
D.1.832 rad

Answer: D

\[ \pi \text{ radians} = 180^{\circ} \]

\[ 105^{\circ} = 105 \times \frac{\pi}{180} = \frac{7\pi}{12} \]

\[ = 1.832 \text{ rad} \]

Since a radian is about \( 57.3^{\circ} \), an angle of \( 105^{\circ} \) should come out a little under 2 radians.

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156.

The length and breadth of a rectangle are in the ratio of 9:5. When the sides of the rectangle are extended on each side by 5 m the ratio of length to breadth becomes 5:3. What is the area of the original rectangle?

A.1160 sq.m
B.1565 sq.m
C.1125 sq.m
D.1045 sq.m

Answer: C

Let the length and breath of rectangle be 'X' m and 'Y' m respectively.
Given, length and breadth of a rectangle are in the ratio of 9 : 5.
X/Y = 9/5
5X = 9Y
5X - 9Y = 0 ....(i)

If sides of the rectangle are extended by 5 m, ratio becomes 5 : 3.
(X + 5)/(Y + 5) = 5/3
3(X + 5) = 5(Y + 5)
3X + 15 = 5Y + 25
3X - 5Y = 10 ....(ii)
By solving equation (i) and (ii),
X = 45 m ; Y = 25 m
So, length of rectangle = 45 m
Breadth of rectangle = 25 m

WKT, Area of rectangle = l x b
Area of rectangle = 45 x 25 = 1125 sq.m.

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