Question

point a is the incenter of triangle def. which must be true? select three options
□ point a is the center of the circle that passes through points d, e, and f.
□ point a is the center of the circle that passes through points x, y, and z.
□ (overline{za} cong overline{ya})
□ (overline{ea} cong overline{fa})
□ (overline{ax} cong overline{ay})

Explanation:

1. Recall the definition of the incenter: The incenter of a triangle is the intersection of the angle bisectors and is equidistant from all sides of the triangle. The circle centered at the incenter and tangent to the sides (passing through the points of tangency \(X\), \(Y\), \(Z\)) is the incircle.
2. Analyze each option:

• "Point A is the center of the circle that passes through points D, E, and F": The circumcenter (not incenter) is the center of the circle passing through the vertices. So this is false.
• "Point A is the center of the circle that passes through points X, Y, and Z": Since \(A\) is the incenter, it is equidistant from the sides (so \(AX = AY = AZ\)), making it the center of the incircle (passing through \(X\), \(Y\), \(Z\)). This is true.
• "\(\overline{ZA} \cong \overline{YA}\)": \(ZA\) and \(YA\) are distances from incenter \(A\) to sides \(EF\) and \(DF\) respectively. By incenter property, they are equal. This is true.
• "\(\overline{EA} \cong \overline{FA}\)": \(EA\) and \(FA\) are segments from vertex \(E\) and \(F\) to incenter \(A\). There's no property stating these are equal (unless the triangle is isoceles in a specific way, which isn't given). So this is false.
• "\(\overline{AX} \cong \overline{AY}\)": \(AX\) and \(AY\) are distances from incenter \(A\) to sides \(DE\) and \(DF\) respectively. By incenter property, they are equal. This is true.

Answer:

B. Point A is the center of the circle that passes through points X, Y, and Z
C. \(\overline{ZA} \cong \overline{YA}\)
E. \(\overline{AX} \cong \overline{AY}\)