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Area Questions and Answers

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131.

Which one of the following has a greater perimeter than the rest?

Answer: C

Work out all four perimeters and compare.

Square, area 36: side \( \sqrt{36} = 6 \), so

\[ P = 4 \times 6 = 24 \text{ cm} \]

Equilateral triangle, side 9:

\[ P = 3 \times 9 = 27 \text{ cm} \]

Rectangle, length 10 and area 40: breadth \( \tfrac{40}{10} = 4 \), so

\[ P = 2(10 + 4) = 28 \text{ cm} \]

Circle, radius 4:

\[ C = 2\pi (4) = 25.13 \text{ cm} \]

The rectangle's 28 cm is the largest. It is also the least compact shape here, which is the general pattern: for a given area, the more elongated the figure, the longer its boundary - and the circle is always the most economical.

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132.

A big rectangular plot of area 4320 Sq.m is divided into 3 square shaped smaller plots by fencing parallel to the smaller side of the plot. However, some area of land was still left as a square plot could not be formed. So 3 more square shaped plots were formed by fencing parallel to the longer side of the original plot. Such that no area of the plot was left a surplus. What are the dimensions of the original plot?

Answer: C

Let the side of smaller square at the extreme right be a 

Then side of the larger square is 3a 

Total area of the field = a² + a² +a² + (3a)² +(3a)² = 30a²

Therefore, 30 a² = 4320 => a² = 432 / 3 = 144 => a = 12 

Smaller side of the field = 36 

 

Larger side of the field = (3a) × 3 + a = 10 a = 10 × 12 = 120 

 

Original Dimensions of the field = 120m × 36m

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133.

In a Circle of Radius 5 cm if AB and AC are two equal chords of length 6cm each, then the length of chord BC is

Answer: D

Place the centre at the origin with \( A = (0,\,5) \). By symmetry, equal chords \( AB \) and \( AC \) put B and C at the same height, so \( BC \) is horizontal.

For \( B = (x,\,y) \) on the circle with \( AB = 6 \):

\[ x^{2} + y^{2} = 25, \qquad x^{2} + (y - 5)^{2} = 36 \]

Subtracting:

\[ -10y + 25 = 11 \quad\Longrightarrow\quad y = 1.4 \]

\[ x^{2} = 25 - 1.96 = 23.04 \quad\Longrightarrow\quad x = 4.8 \]

So \( B = (4.8,\,1.4) \) and \( C = (-4.8,\,1.4) \):

\[ BC = 2 \times 4.8 = 9.6 = \frac{48}{5} \text{ cm} \]

None of the listed lengths is \( \tfrac{48}{5} \), so the answer is none of these. The \( \tfrac{24}{5} \) on offer is only half of \( BC \) - the distance from B to the axis of symmetry.

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134.

If the length of diagonal AC of a square ABCD is 5.2 cm then area of the square ABCD is:

Answer: B

A square's diagonal and side satisfy \( d = a\sqrt{2} \), so the area follows from the diagonal alone.

\[ A = a^{2} = \frac{d^{2}}{2} \]

\[ A = \frac{(5.2)^{2}}{2} = \frac{27.04}{2} = 13.52 \text{ cm}^{2} \]

The diagonal cuts the square into two congruent right triangles, each with legs \( a \) and hypotenuse \( d \) - which is where the halving comes from. Forgetting it gives 27.04.

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