McGraw Hill Glencoe Geometry, 2012
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McGraw Hill Glencoe Geometry, 2012 View details
1. Bisectors of Triangles
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Exercise 8 Page 329

Using the Incenter Theorem, find the lengths of the legs in triangle

Practice makes perfect

In this exercise we are asked to find the measure of Let's find this segment on the given diagram.

From the diagram we can see that and are perpendicular to the sides of the triangle. This means that these segments are the distances from to the sides of Also, we are told that is the incenter of Thus, we can use the Incenter Theorem.

The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle.

According to the theorem, the incenter is equidistant from the sides of the triangle This means that distances and are the same.
We also know that the measure of is Therefore, the measure of is too. Now, let's consider the triangle
is a right triangle, because is perpendicular to Thus, we can use the Pythagorean Theorem.
In our case, the legs of the triangle are and and the hypotenuse is so this theorem can be written the following way.
We know the length of the two triangle's legs. Let's substitute for and for and solve the equation we will get for
The measure of is about