Question

which of the following triangles indicates the incenter at point z? *

Explanation:

The incenter of a triangle is the intersection point of the angle bisectors, and it is also the center of the incircle (a circle tangent to all three sides of the triangle). Let's analyze each option:

• Option I: The dashed lines with right angles suggest perpendicular bisectors (for circumcenter) or altitudes, not angle bisectors. So this is not the incenter.
• Option J: There is a circle (incircle) tangent to all three sides of the triangle, and point \( Z \) is at the center of this incircle. This matches the definition of the incenter (center of the incircle, intersection of angle bisectors).
• Option K: The triangle is inscribed in a circle (circumcircle), and the right angles on the dashed lines suggest perpendicular bisectors (circumcenter), not incenter.
• Option L: The marks on the sides suggest congruent segments (midsegments or perpendicular bisectors for circumcenter/centroid), not angle bisectors. So this is not the incenter.

Answer:

J. The triangle with the incircle centered at \( Z \) (option J) indicates the incenter at point \( Z \) because the incenter is the center of the incircle (tangent to all three sides), and this figure shows a circle tangent to all sides with \( Z \) at its center.