Question

1. which point is equidistant from all sides of a triangle? a. incenter b. tangent point c. centroid d. circumcenter 8. what is the key property of the perpendicular bisectors in a triangle? a. they always intersect at the centroid. b. they are always parallel to the sides of the triangle. c. they divide each side into equal halves. d. they are equidistant from all sides of the triangle. 9. if the central angle of a sector is 2π/3 radians and the radius is 8 meters, what is the sector area? a. 64π/3 square meters b. 32π/3 square meters c. 52π square meters d. 16π square meters

Explanation:

1. The in - center of a triangle is the point of intersection of the angle bisectors of the triangle and is equidistant from all sides of the triangle.
2. Perpendicular bisectors of a triangle are lines that are perpendicular to the sides of the triangle and divide each side into two equal halves.
3. The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$, where $r$ is the radius and $\theta$ is the central angle in radians. Given $r = 8$ meters and $\theta=\frac{2\pi}{3}$, we have $A=\frac{1}{2}\times8^{2}\times\frac{2\pi}{3}=\frac{1}{2}\times64\times\frac{2\pi}{3}=\frac{64\pi}{3}$ square meters.

Answer:

1. a. Incenter
2. c. They divide each side into equal halves.
3. a. $\frac{64\pi}{3}$ square meters