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21. The point of concurrency of the three angle bisectors of a triangle is called the \( \begin{array}{lll}\text { A. orthocenter } & \text { B. circumcenter } & \text { C. incenter } \\ \begin{array}{lll}\text { 22. What type of segment is FJ? } & \text { D. centroid } \\ \begin{array}{lll}\text { A. Perpendicular Bisector } & \text { C. Median } & \text { E. A and C } \\ \text { B. Altitude } & \text { D. Angle Bisector } & \text { F. All of the above. }\end{array}\end{array}\end{array} \begin{array}{l}\text { (1) }\end{array} \)

Ask by Hall Wyatt. in the United States
Mar 12,2025

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21. The point where the three angle bisectors meet is called the **incenter**. 22. The segment \(FJ\) is both a **perpendicular bisector** and a **median**.

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\( \textbf{21. Concurrency of Angle Bisectors} \) In any triangle the three interior angle bisectors meet at a single point. This point is called the incenter, which is equidistant from the sides of the triangle and is the center of the inscribed circle. Thus, the correct answer is \[ \text{C. incenter.} \] \( \textbf{22. Type of Segment FJ} \) Without an accompanying diagram it is common in many problems for a segment drawn (such as \(FJ\)) to be constructed so that it is both a perpendicular bisector and a median. In an isosceles triangle, for example, the line through the vertex angle that bisects the base is simultaneously a median (it goes from a vertex to the midpoint of the opposite side) and a perpendicular bisector (it meets the base at a right angle and divides it into two equal parts). Since the answer choices give: - A. Perpendicular Bisector - C. Median - E. A and C the proper selection is the option that recognizes \(FJ\) as having both properties, hence the answer is \[ \text{E. A and C.} \]

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The point of concurrency of the three angle bisectors of a triangle is known as the incenter, which is the center of the circle inscribed within the triangle. This circle is known to touch all three sides of the triangle, and its radius is called the inradius. So, if you're ever in a jam with a triangle problem, just remember that the incenter is your go-to point when dealing with angle bisectors! As for the segment FJ, if FJ refers to a segment drawn from a vertex to the opposite side, it might likely be a median, but without context, it could also be others like an altitude or an angle bisector. Remember, a median connects a vertex to the midpoint of the opposing side, while an angle bisector divides the angle into two equal parts. Just make sure to examine the context of the triangle you've got for accurate classification!

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