The heart-shaped decoration shown in the figure above consists of a square and two semicircles...

[AAPL]

Official Guide

The heart-shaped decoration shown in the figure above consists of a square and two semicircles. What is the radius of each semicircle?

1) The diagonal of the square is \(10\sqrt{2}\) centimeters long.
2) The area of the square region minus the sum of the areas of the semicircular regions is \(100 - 25\pi\) square centimeters.

OA D

[Ignite]

We are required to find the radius of each semicircle.

Observing the figure, we know that if we know the measurement of the side of the square, we can find the radius of semicircles.
i.e. If the side of the square is 'a', the radius of the semi-circle is \(\frac{a}{2}\)

1. We have the diagonal of the square as \(10\sqrt{2}\).
If the side of a square is 'a', the diagonal of the square would be \(a\sqrt{2}\).

Thus, we get that a = 10.
Hence, radius = 5

Sufficient.


2. If the side of the square is 'a'
Area of the square = \(a^2\) -> (a)
Area of each semicircle = \(\frac{1}{2}\cdot\pi\cdot\left(\frac{a}{2}\right)^2\)
Then sum of areas of semicircle = \(\pi\cdot\left(\frac{a}{2}\right)^2\) ->(b)

Now given that,
(a) - (b) = \(100-25\pi\)

Equating both sides we get a = 10


Hence, radius - 5

Sufficient.

Answer D