# Math Models

## Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem


### 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  is called 
The three angles of a triangle are related in a special way. The sum of their measures is  Note that we read  as “the measure of angle A.” So in  in ,
Because the perimeter of a figure is the length of its boundary, the perimeter of  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  angle with the base. We will draw  again, and now show the height, h. See .
 
The triangle properties we used so far apply to all triangles. Now we will look at one specific type of triangle—a right triangle. A right triangle has one  angle, which we usually mark with a small square in the corner.
In the examples we have seen so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.

### Use the Pythagorean Theorem

We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the Pythagorean Theorem. This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.
Before we state the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a  angle, marked with a small square in the corner. The side of the triangle opposite the  angle is called the hypotenuse and each of the other sides are called legs.
The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. In symbols we say: in any right triangle,  where  are the lengths of the legs and  is the length of the hypotenuse.
Writing the formula in every exercise and saying it aloud as you write it, may help you remember the Pythagorean Theorem.
To solve exercises that use the Pythagorean Theorem, we will need to find square roots. We have used the notation  and the definition:
If  then  for 
For example, we found that  is 5 because 
Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.

### Solve Applications Using Rectangle Properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right  angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L, and its adjacent side as the width, W.
The distance around this rectangle is  or  This is the perimeter, P, of the rectangle.
What about the area of a rectangle? Imagine a rectangular rug that is 2-feet long by 3-feet wide. Its area is 6 square feet. There are six squares in the figure.
The area is the length times the width.
The formula for the area of a rectangle is 
 
We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

### Key Concepts

1. Problem-Solving Strategy for Geometry Applications


2. Triangle Properties For 
Angle measures:


Perimeter:


Area:


A right triangle has one  angle.
3. The Pythagorean Theorem In any right triangle,  where c is the length of the hypotenuse and a and b are the lengths of the legs.
4. Properties of Rectangles



### Practice Makes Perfect

Solving Applications Using Triangle Properties
In the following exercises, solve using triangle properties.
Use the Pythagorean Theorem
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth, if necessary.
In the following exercises, solve using the Pythagorean Theorem. Approximate to the nearest tenth, if necessary.
Solve Applications Using Rectangle Properties
In the following exercises, solve using rectangle properties.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?