Solved on Nov 24, 2023
Choose 3 true statements: 1) The centroid is the concurrency of a triangle's altitudes. 2) The incenter is the concurrency of a triangle's angle bisectors. 3) The circumcenter is the concurrency of a triangle's perpendicular bisectors and is equidistant from the vertices.
STEP 1
Assumptions1. We understand the definitions of the following geometric terms point of concurrency, altitude, angle bisector, perpendicular bisector, median, centroid, circumcenter, orthocenter. . We understand the properties of these points of concurrency in relation to a triangle.
STEP 2
Let's evaluate each statement one by one.
The first statement says "The point of concurrency of the three altitudes of a triangle is called the centroid."
This statement is incorrect. The point of concurrency of the three altitudes of a triangle is called the orthocenter, not the centroid.
STEP 3
The second statement says "The point of concurrency of the three angle bisectors of a triangle can be used to inscribe a circle inside a triangle."
This statement is correct. The point of concurrency of the three angle bisectors of a triangle is called the incenter, and it is the center of the inscribed circle (incircle) of the triangle.
STEP 4
The third statement says "The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter."
This statement is correct. The point of concurrency of the three perpendicular bisectors of a triangle is indeed called the circumcenter.
STEP 5
The fourth statement says "The point of concurrency of the three medians of a triangle is called the orthocenter."
This statement is incorrect. The point of concurrency of the three medians of a triangle is called the centroid, not the orthocenter.
STEP 6
The fifth statement says "The circumcenter is equidistant from the three vertices of the triangle."
This statement is correct. The circumcenter is the center of the circumscribed circle (circumcircle) of the triangle, which means it is equidistant from the three vertices of the triangle.
STEP 7
The sixth statement says "Perpendicular bisectors must include a vertex of the triangle."
This statement is incorrect. Perpendicular bisectors of a triangle are lines, segments, or rays that divide each side of the triangle into two equal parts at a90° angle, and they do not necessarily pass through a vertex of the triangle.
So, the three true statements are- The point of concurrency of the three angle bisectors of a triangle can be used to inscribe a circle inside a triangle.
- The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter.
- The circumcenter is equidistant from the three vertices of the triangle.
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