Question

I don’t understand how to do this problem

Answer 1

First find the area of the sector, which includes the shaded region's area and the area of the triangle JKL. Let $A$ be the area of the sector. This area occurs in a fixed ratio with the area of the entire circle based on the measure of the central angle subtended by the arc LK:

$\dfrac A{54^\circ}=\dfrac{27^2\cdot3.14}{360^\circ}\implies A\approx343.359$

To get the area of the shaded region, subtract from $A$ the area of the triangle.

The area of a triangle is 1/2 the base times the height. If we bisect the central angle with a line segment that meets the side KL at its midpoint, then we get a right triangle with hypotenuse 27 and one angle of measure 27º (half of the central angle). This triangle has base $b$ and height $h$ such that

$\sin27^\circ=\dfrac b{27}\implies b\approx12.258$

$\cos27^\circ=\dfrac h{27}\implies h\approx24.057$

So the right triangle has area approximately 1/2*12.258*24.057, or about 147.443. Triangle JKL is made up of two of these right triangles, so it has area of 294.887.

Subtracting this from $A$ gives an area of the shaded region of about 48.472, which we round up to 48.5.