Question

how would you start to find the incenter of δabc? a draw a line parallel to one side that divides the other two sides proportionally. b determine if the combined length of any two sides is greater than the length of the third side. c determine the measure of each interior angle and confirm that the sum of the angles equal 180°. d construct two angle bisectors that divides the opposite sides into two proportional segments.

Explanation:

The in - center of a triangle is the point of intersection of the angle bisectors of the triangle. To find the in - center, we first need to construct the angle bisectors.

• Option A: Drawing a line parallel to one side that divides the other two sides proportionally is related to the basic proportionality theorem (Thales' theorem) and has nothing to do with finding the in - center.
• Option B: Determining if the combined length of any two sides is greater than the length of the third side is the triangle inequality theorem, which is used to check if a triangle can be formed, not for finding the in - center.
• Option C: Determining the measure of each interior angle and confirming that the sum of the angles equal \(180^{\circ}\) is a property of triangles (angle - sum property) and not related to finding the in - center.
• Option D: Constructing two angle bisectors (since the in - center is the intersection of angle bisectors) is the correct first step. The angle bisector theorem states that an angle bisector of a triangle divides the opposite side into segments proportional to the adjacent sides.

Answer:

D. Construct two angle bisectors that divides the opposite sides into two proportional segments.