Question

1. what is the first step to construct the centroid of a triangle? a. construct the perpendicular bisectors of each side b. draw a circle in the triangle c. construct the angle bisectors of each angle d. identify the midpoints of each side 2. if a circles sector has an area of 25π square cm and the central angle is π/2 radians, what is the radius? a. 5 cm b. 20 cm c. 10 cm d. 12 cm 3. what is the measure of an angle inscribed in a semi - circle? a. 90° b. 60° c. 180° d. 45°

Explanation:

Step1: Recall centroid - construction

The centroid of a triangle is the point of intersection of its medians. A median is a line - segment joining a vertex to the mid - point of the opposite side. So the first step is to identify the mid - points of each side.

Step2: Recall sector - area formula

The area formula for a circle sector is $A=\frac{1}{2}r^{2}\theta$, where $A$ is the area of the sector, $r$ is the radius and $\theta$ is the central angle in radians. Given $A = 25\pi$ and $\theta=\frac{\pi}{2}$, we substitute into the formula: $25\pi=\frac{1}{2}r^{2}\frac{\pi}{2}$. Cross - multiply to get $100\pi=r^{2}\pi$, then $r^{2} = 100$, so $r = 10$ cm.

Step3: Recall inscribed - angle theorem

The measure of an inscribed angle in a semi - circle is always $90^{\circ}$ according to the inscribed - angle theorem.

Answer:

1. d. Identify the midpoints of each side
2. c. 10 cm
3. a. $90^{\circ}$