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Geometry Honors Chapter 5 Constructions Worksheet Name _________________________________ Honor Code: ______________________________________________________________________________________ 1. Construct the perpendicular bisectors of each of the following triangles. Label the point of concurrency of the perpendicular bisectors P and answer the following questions. Acute Triangle Right Triangle -1- Obtuse Triangle Properties of the perpendicular bisector: A perpendicular bisector is a segment, ray, line, or plane that is perpendicular to a segment at its _____________. The point of concurrency (intersection) of a triangle's perpendicular bisectors is called the ___________________. The circumcenter is equidistant from all three ___________________ of the triangle. The circumcenter is the center of the ____________________ circle, which can be constructed through all three vertices. In a right triangle, the circumcenter is also the ___________________ of the hypotenuse. In an acute triangle, the circumcenter is located __________________ the triangle. In an obtuse triangle, the circumcenter is located __________________ the triangle. -2- 2. Construct the angle bisectors of each of the following triangles. Label the point of concurrency of the angle bisectors P and answer the following questions. Acute Triangle Right Triangle -3- Obtuse Triangle Properties of the angle bisector: An angle bisector _____________ a vertex angle. The point of concurrency (intersection) of the three angle bisectors is called the ______________________. The incenter is equidistant from the three _________________ of the triangle. The incenter is the center of the circle that can be _____________________ inside the triangle. The incenter is always located ____________________ of the triangle. An angle bisector will intersect the opposite side, but it may or may not go through the midpoint. -4- 3. Construct the medians of each of the following triangles. Label the point of concurrency of the medians P and answer the following questions. Acute Triangle Right Triangle -5- Obtuse Triangle Properties of the median: A median passes through a vertex of the triangle and the __________________ of the opposite side. The intersection of the three medians is called the __________________. The distance from the vertex to the centroid is ______________ the length of the median. The distance from the centroid to the side midpoint is ____________ the length of the median. The centroid is the center of ___________________ of a triangle. In acute, right, and obtuse triangles, the centroid is located _________________ the triangle. A median passes through a vertex, but may or may not bisect the angle. -6- 4. Construct the altitudes of each of the following triangles. Label the point of concurrency of the altitudes P and answer the following questions. Acute Triangle Right Triangle -7- Obtuse Triangle Properties of the altitude: An altitude passes through a vertex and is ___________________ to the opposite side. The intersection of the altitudes is called the _______________________ In an acute triangle, the orthocenter is located _______________ the triangle. In an obtuse triangle, the orthocenter and two of the altitudes are located __________________ of the triangle. In a right triangle, the orthocenter falls on the _____________________ of the right angle and the two legs are also altitudes. -8-