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Comparing the Length of the Altitude, Angle Bisector, and Median in a Triangle                 Searching for

Median, Altitude, and Bisector

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Lesson Focus

Comparing the Length of the Altitude, Angle Bisector, and Median in a Triangle

Geometry

Students determine the relationship between the lengths of the altitude, angle bisector, and median drawn from the same vertex of the triangle.

Now You Know

After completing this tutorial, you will be able to complete the following:

• Identify the altitude, angle bisector, and median by using the relationship between their lengths.
• Find the length of the altitude, angle bisector, and median by using the coordinates of their endpoints and the distance formula.

Everything You'll Have Covered

Recall that the altitude is the perpendicular line segment from a vertex to the line containing the opposite side. The median of a triangle is a line segment from a vertex of a triangle to the midpoint of the opposite side. An angle bisector of a triangle is a line or line segment that divides an angle of the triangle into two equal parts. When we refer to the angle bisector line segment, as we do in this Activity Object, we mean the line segment from a vertex of a triangle to the opposite side that bisects the vertex angle. Any triangle has three altitudes, three medians, and three angle bisectors.

In an isosceles triangle, the altitude drawn from the angle whose sides are the congruent sides of the triangle will divide the triangle into two smaller congruent triangles. Indeed, Thus, by angle-angle-side congruency, Then, since congruent parts of congruent triangles are congruent, Therefore, D is the midpoint of BC , and AD is a line segment whose endpoints are a vertex and the midpoint of the opposite side. As a result, AD is the median drawn from vertex A.

Again, since congruent parts of congruent triangles are congruent, This means that AD bisects , and is therefore the angle bisector from A as well. This means that in an isosceles triangle, if we consider the vertex whose sides are congruent sides of the triangles, the altitude, angle bisector, and median drawn from that vertex have the same length. Since equilateral triangles are isosceles, this holds for equilateral triangles as well. However, since all sides of an equilateral triangle are isosceles, this means that all of the altitudes, angle bisectors, and medians have the same length.

We can also find a relationship between the lengths of the line segments in scalene triangles. In general, we have the following relationship between the lengths of the line segments drawn from the same vertex: Tutorial Details

 Approximate Time 20 Minutes Pre-requisite Concepts Students should be able to define altitude, median, and angle bisector; use the notation for the length of a line segment; and use the distance formula. Course Geometry Type of Tutorial Skills Application Key Vocabulary altitude, angle bisector, median