# Problems with Bisectors, Altitudes and Medians

## Geometry Quiz

The bisector of an angle is the line or line segment that divides the angle into two equal parts.

The median of a triangle is a line segment that extends from one vertex of a triangle to the midpoint of the opposite side.

An altitude is a segment from the vertex of a triangle to the opposite side that is perpendicular to that segment.

Question 1: The center O of the circle inscribed in any triangle is the point of intersection of the:

Question 2: In triangle ΔABC , point F is the point of intersection of the bisector of angle ∠ABC and the altitude AD of the triangle. The measure of ∠BCA is 48°, and the measure of ∠BAC is 78°. What is the measure of ∠AFE?

Question 3: Calculate the length of the median of an equilateral triangle with sides equal to

Question 4: The median m divides the ABC triangle in 2 triangles, ABM and ACM. What is the ratio between the areas of the 2 triangles, AreaABM/AreaACM ?

Question 5: Triangle ABC in the picture below has ∠ABC = 60o and ∠BCA = 40o. If AD, BE and CF are the bisectors of the triangle, which of the following pairs of triangles are similar:

Question 6: In the picture given below, D is a point on the side BC of triangle ABC. Determine which of the following given assumptions are enough to prove that the ABD and ACD triangles are similar.

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