Question

Find the area of the shaded region. Round your answer to the nearest hundredth. A circle and two triangles drawn inside it. The two circles are congruent and share a common side. This common side is the diameter of circle. The third vertex of both triangles lie on the circle. The lengths of sides apart from diameter are labeled 3 meters and 4 meters. The region inside circle and outside triangles is shaded. The area is about __square meters.

Answer 1

Answer: 7.635 square meters

Explanation:

Both triangles have the diameter as one of their sides and they both have a vertex on the circumference of the circle. Thus, the two triangles are right triangles, and the hypotenuse of the triangles is the diameter.

Use the Pythagorean Theorem:
`a^2+b^2=c^2`

Square root both sides to isolate c (hypotenuse/diameter):
`c=√(a^2+b^2)`

Plug in the values of a and b to calculate c:
`c=√(3^2+4^2)=√(9+16)=√(25)=5`

The diameter of the circle is 5 meters, so the radius of the circle is 2.5 meters.

Plug in the radius into the equation for the area of a circle: Area =
`\pi r^2=\pi *2.5^2=6.25\pi` square meters.

The equation for the area of a triangle is
`(1)/(2)bh`, where b is the base of the triangle and h is the height of the triangle.

Since we have two congruent triangles, the total area of the two triangles combined is simply b*h.

Plug in the values of b and h to get b*h = 3*4 = 12 square meters.

Subtract the total area of the two triangles combined from the area of the circle to get the area of the shaded region:
`6.25\pi - 12=7.635` square meters.