Students define the angle bisectors and incenter of a triangle, explore the properties of angle bisectors, and use these properties to solve problems.
After completing this tutorial, you will be able to complete the following:
Consider the following problem: A soccer goalie's position relative to the ball and goalposts forms congruent angles. Will the goalie have to move closer to the left or right goalpost in order to block a shot?
We can solve this problem using angle bisectors. An angle bisector of a triangle is a line or line segment that divides an angle of the triangle into two equal parts. Sometimes, an angle bisector is called an interior angle bisector, since it bisects an interior angle of the triangle. Any triangle has three interior angle bisectors corresponding to each of the triangle's interior angles. We have the following theorem about the concurrency of the angle bisectors of a triangle.
Theorem 1: The bisectors of the angles of a triangle intersect at a point that is equidistant from the three sides of the triangle.
Proof: Consider triangle ABC. Suppose the bisectors of angles A and C intersect at point P.
First construct perpendicular segments from P to BC, AC, and AB, and denote the points of intersection as X, Y, and Z, respectively.Since AP bisects angle A, So, by angle-angle-side congruency. Since corresponding parts of congruent triangles are congruent, this means that Similarly, by angle-angle-side congruency and Thus, point P is equidistant from the sides of the triangle. Finally,because their legs and hypotenuses are congruent, and so
Thus, the angle bisectors of a triangle intersect at a point that is equidistant from the three sides of the triangle.
The point of concurrency of the angle bisectors is called the incenter of the triangle, and it always lies inside the triangle. Since the incenter is equidistant from the sides of the triangle, it is the center of a circle inside the triangle. In fact, the incenter of a triangle is the center of the largest circle contained inside the triangle, known as the inscribed circle of the triangle (see Figure 2).
For this reason, the incenter (and angle bisectors) has many real-life applications. For example, the largest circular pool contained in a triangle-shaped backyard will have its center at the incenter of the triangle. The incenter is equidistant from all three sides of the triangle. However, angle bisectors can be used to find points equidistant from two sides as well.
Theorem 2: A point lies on the bisector of an angle if and only if it is equidistant from the sides of the angle.
Proof: First, consider with K a point on angle bisector AD. Draw perpendicular segmentsKT and KS to AB and AC, respectively (See Figure 3). By angle-angle-side congruency, Since corresponding parts of congruent triangles are congruent, and so K is equidistant to AB and AC.
On the other hand, suppose that D is a point inside angle A, which is equidistant to both AB and AC. Then we can draw perpendicular, congruent line segments from Z to the sides. Let P and Q be the points of intersection of these line segments with AB and AC. respectively. Thus, by the Pythagorean theorem. Since corresponding parts of congruent triangles are congruent, and so D lies on the angle bisector of angle A.
We can represent the soccer problem as follows: Suppose the ball is located at point A, the goalie is at point G, and the left and right end posts are points L and R, respectively.
Since the goalie's position relative to the ball and the goalposts form congruent angles, this means that so the goalie is located on the bisector of Therefore, by Theorem 2, he is equidistant from AR and AL. Notice that the only way for the opposite team to score a point is for the ball to be shot inside angles LAG or RAG. Therefore, the goalie must move the same distance to block a shot towards the left goalpost as he must move to block as shot towards the right goalpost.
Corollaries of Theorem 2 follow. They can be easily proven using Theorem 2 and the Pythagorean theorem:
Corollary 1: In an isosceles triangle, the bisector of the angle adjacent to the two congruent sides is a perpendicular bisector of the remaining side.
Corollary 2: In an equilateral triangle, all angle bisectors are perpendicular bisectors of the opposite sides.
In addition to the above theorems and corollaries, we also have the angle bisector theorem, which relates the lengths of line segments formed by the intersection of an angle bisector and the side opposite the angle of a triangle to the lengths of the other two sides of the triangle.
Interior angle bisector theorem: In triangle ABC, if the angle bisector of angle A intersects side BC at a point D, then the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC. In other words,
Proof: Let ABC be a triangle and D be the point at which the angle bisector of angle A intersects BC. Draw a line through C parallel to AB and extend AD to intersect the line. Let E be the intersection point of the line AD and the line through C.
Because they are vertical angles, Line BC forms a transversal of parallel lines AB and CE. Thus and are alternate exterior angles and are therefore congruent. Similarly, AE is a transversal of AB and CE so alternate exterior angles DEC and DAB are also congruent. Therefore, is similar to and so Also Note that is isosceles because it has two congruent angles. Therefore, we can substitute so that as required.
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should know the definitions of triangle, bisector, tangent point, angle bisector, circle, and radius; and understand the concepts of the Pythagorean theorem, ratio and proportion, and alternate exterior angles.|
|Type of Tutorial||Concept Development|
|Key Vocabulary||angle bisector, incenter of a triangle, interior angle bisector|