Concurrency Properties of Triangles and Triangle Thoerems HW
Matching:
A. Centroid ______the point where all the altitudes meet
B. Orthocenter ______the point where all the medians meet
C. Circumcenter ______the point where all the angle bisectors meet
D. Incenter ______the point where all the perpendicular bisectors of the sides meet
I) Centroid:
Theorem: The medians of a triangle are concurrent and intersect each other in a ratio of 2:1.
1. If point R is the centroid of triangle ABC.
What is the perimeter of triangle ABC given
that segments CF, DB, and AE are equal to 2, 3
and 4 respectively?
Perimeter of triangle ABC = ______
2. Point Z is the centroid of triangle ABC and CZ = 18.
What is the length of segment ?
II) Circumcenter:
Theorem: Perpendicular bisectors of sides of a triangle are concurrent at a point equidistant from the vertices.
HK and JK are angle bisectors of ∆HIJ. Find each measure.
1. The distance from K to JI ______
2. m < HJK ______
3. m < JKH ______
4. m < HJI ______
III) Incenter:
Theorem: The bisectors of the angles of a triangle meet at a point that is equally distant from the sides of the triangle.
1. Which of the following points in the
diagram below is the incenter of triangle ABC?
How do you know?
2. Given that point S is the incenter of right triangle PQR and angle RQS is 30°, what are the measures of anglesRSQ and RPQ?
mÐRSQ = _______ mÐRPQ = ______
IV) Orthocenter:
Theorem: The point where the lines containing the altitudes are concurrent is called the orthocenter of a triangle.
1. Which point is the intersection of the altitudes of a triangle?
a. orthocenter b. centroid c. incenter d. circumcenter
2. Which type of triangle would have its orthocenter on the triangle?
a. right b. obtuse c. acute d. equilateral
3. Which type of triangle would have its orthocenter inside the triangle?
a. right b. obtuse c. acute d. equilateral
4. Which type of triangle would have its orthocenter outside the triangle?
a. right b. obtuse c. scalene d. equilateral