Answer:
Hence, Area of shaded region is:
(4π-8) cm^2
Step-by-step explanation:
We are asked to find the area of the shaded region.
we could clearly observe that:
Area of shaded region=Area of semicircle-Area of right triangle.
As we are given two angles of a triangle so the third angle is equal to 45°.
We could find the length of the side with the help of the trignometric formula as:
tan 45=MN/LM
1=MN/4
MN=4 cm
Similarly we can find the side LN with the help of the Pythagorean theorem as:
[tex]LN^2=4^2+4^2\\\\LN^2=16+16\\\\LN^2=32\\\\LN=\sqrt{32}\\\\LN=4\sqrt{2}[/tex]
Hence, Area of triangle is:
[tex]\dfrac{1}{2}\times {MN}\times {LM}\\\\=\dfrac{1}{2}\times 4\times 4\\\\=8 cm^2[/tex]
similarly we have diameter of semicircle=4√2 cm.
Hence, Radius of circle=2√2 cm.
Area of semicircle is given by:
[tex]\dfrac{\pi r^2}{2}\\\\=\dfrac{1}{2}\times \pi\times 2\sqrt{2}\times 2\sqrt{2}\\\\=4\pi cm^2[/tex]
Hence, Area of shaded region is:
(4π-8) cm^2
