Question

1. choose the correct answer.

which triangle illustrates an incenter?

Explanation:

The incenter of a triangle is the point where the angle bisectors intersect, and it is equidistant from all sides (the distances to each side are equal, represented by perpendicular segments from the incenter to the sides). Let's analyze each option:

• Option A: Shows segments related to medians (since \( X \) is the midpoint of \( AC \), \( Y \) midpoint of \( AB \), \( Z \) midpoint of \( BC \)), so it's the centroid (intersection of medians), not incenter.
• Option B: The segments from the point to the sides are perpendicular (right angles at \( Y, Z, \) and the base), and the angle bisectors (the lines from vertices \( A, B, C \) seem to bisect angles, as seen by the equal angle markings at \( B \) and equal segments from \( A \) to the sides). The incenter is equidistant from all sides, and here the perpendicular distances (like \( YP, ZP, \) and the one to \( AC \)) should be equal, which matches the incenter's property.
• Option C: The perpendicular segments from \( P \) to \( AB, BC, \) and the median - like segment to \( AC \) don't indicate equal perpendicular distances (or angle bisectors), more like a circumcenter or other center.
• Option D: The equal segments \( XP = BP=CP \) suggest it's the circumcenter (equidistant from vertices), not incenter.

So the triangle with the incenter is Option B.

Answer:

B. The triangle in option B (where the point is equidistant from all sides via perpendicular segments and lies at the intersection of angle bisectors)