Question

Find the area of the shaded region

Answer 1

4pi - 4 or 8.57

Explanation:

First, we can use the 45, 45, 90 triangle to establish that the length of line LN is 4*sqrt(2). This means that the radius of the circle is 2*sqrt(2). The area of the circle is then pi*r*r which is 8pi.

The radius of the circle, or Line LO, is 2*sqrt(2). If we divide the triangle in half, we can solve that line MO is also 2*sqrt(2). Using the formula b*h*1/2, we can derive that the area of the half triangle LOM is 4. Doubling that value tells us that the are of LMN is 8.

Finally we can subtract the area of the half circle - area of the triangle which is 4pi - 4.

Answer 2

Explanation:

∠LMN=90°

∠MLN=45°

So ∠MNL=45°

MN=ML=4 cm (in a triangle sides opposite to equal angles are equal.)

LN=√(4²+4²)=4√2 cm

radius r=1/2 LN=2√2 cm

area of semicircle=1/2 ×π×r²=1/2 ×π×(2√2)²=4 π≈4×3.14=12.56 cm²

area of triangle=1/2×4×4=8 cm²

area of shaded region≈12.56-8≈4.56 cm²