3.4 Solve geometry applications: triangles, rectangles, and the

Solve applications using properties of triangles

In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.

We will start geometry applications by looking at the properties of triangles. Let’s review some basic facts about triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

The plural of the word vertex is vertices . All triangles have three vertices . Triangles are named by their vertices: The triangle in [link] is called $\text{△}ABC.$

The three angles of a triangle are related in a special way. The sum of their measures is $180\text{°}.$ Note that we read $m\text{∠}A$ as “the measure of angle A.” So in $\text{△}ABC$ in [link] ,

Because the perimeter of a figure is the length of its boundary, the perimeter of $\text{△}ABC$ is the sum of the lengths of its three sides.

To find the area of a triangle, we need to know its base and height. The height is a line that connects the base to the opposite vertex and makes a $90\text{°}$ angle with the base. We will draw $\text{△}ABC$ again, and now show the height, h . See [link] .

Triangle properties

For $\text{△}ABC$

Angle measures:

$m\text{∠}A+m\text{∠}B+m\text{∠}C=180$

• The sum of the measures of the angles of a triangle is $180\text{°}.$

Perimeter:

$P=a+b+c$

• The perimeter is the sum of the lengths of the sides of the triangle.

Area:

$A=\frac{1}{2}bh,b=\text{base},h=\text{height}$

• The area of a triangle is one-half the base times the height.