The figure shows a quarter circle in a square. Its centre is at one vertex, and it passes through two Class 9

The figure shows a quarter circle in a square. Its centre is at one vertex, and it passes through two Class 9

Question 1.

The figure shows a quarter circle in a square. Its centre is at one vertex, and it passes through two adjacent vertices. There are two semicircles on two adjacent sides as diameters. They create the shaded regions A and B. Class 9

Show that A and B have equal area.

Solution:

From the figure,

Let side of square = a units.

∴ Radius of quarter circle = a units

Radius of each of two semi circle = $\frac{a}{2}$ units.

∵ Area of quarter circle = $\frac{1}{4}$ πa² sq. units ... (i)

∵ Area of each semicircle

$\begin{aligned} & =\frac{1}{2} \pi\left(\frac{a}{2}\right)^2 \\ & =\frac{1}{8} \pi a^2 \text { sq. units }\end{aligned}$

∴ Area of two semicircles = $\frac{1}{4}$ πa² ... (ii)

∵ Area of quarter circle = Area of two semicircles ...(iii) [from (i) and (ii)]

Now,

Let W be common white area inside the square covered by both semi-circle and quarter circle.

From the figure

Area of two semi-circle

= 2 × Area (W) + 2 × Area (A) ...(iv)

Area of Quarter-circle

= 2 × Area (W) + Area (B) + Area (A) ...(v)

From (iii), (iv) and (v)

2 × Area (W) + 2 × Area (A)

= 2 × Area (W) + Area (B) + Area (A)

∴ Area (A) = Area (B)