Geometry For Dummies
Book image
Explore Book Buy On Amazon

The Angle-Bisector theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the other two sides. The following figure illustrates this.

image0.jpg

The Angle-Bisector theorem involves a proportion — like with similar triangles. But note that you never get similar triangles when you bisect an angle of a triangle (unless you bisect the vertex angle of an isosceles triangle, in which case the angle bisector divides the triangle into two congruent triangles).

Don’t forget the Angle-Bisector Theorem. (For some reason, students often do forget this theorem.) So whenever you see a triangle with one of its angles bisected, consider using the theorem.

How about an angle-bisector problem? Why? Oh, just BCUZ.

image1.jpg

Given: Diagram as shown

Find: 1.) BZ, CU, UZ, and BU and 2.) The area of triangle BCU and triangle BUZ

  1. Find BZ, CU, UZ, and BU.

    image2.png

    It’s a 6-8-10 triangle, so BZ is 10.

    Next, set CU equal to x. UZ then becomes 8 – x. Set up the angle-bisector proportion and solve for x:

    image3.png

    So CU is 3 and UZ is 5.

    The Pythagorean Theorem then gives you BU:

    image4.png
  2. Calculate the area of triangle BCU and triangle BUZ.

    Both triangles have a height of 6 (when you use segment CU and segment UZ as their bases), so just use the triangle area formula:

    image5.png

Note that the ratio of the areas of these triangles, 9 : 15 (which reduces to 3 : 5), is equal to the ratio of the triangles’ bases, 3 : 5. This equality holds whenever a triangle is divided into two triangles with a segment from one of its vertices to the opposite side (whether or not this segment cuts the vertex angle exactly in half).

About This Article

This article is from the book:

About the book author:

Mark Ryan has taught pre-algebra through calculus for more than 25 years. In 1997, he founded The Math Center in Winnetka, Illinois, where he teaches junior high and high school mathematics courses and standardized test prep classes. He also does extensive one-on-one tutoring. He is a member of the Authors Guild and the National Council of Teachers of Mathematics. Ryan is the author of Calculus For Dummies, Calculus Workbook For Dummies, and Geometry Workbook For Dummies.

This article can be found in the category: