Mathematics » Math Models and Geometry I » Use Properties of Rectangles; Triangles; and Trapezoids

Using the Properties of Triangles

Using the Properties of Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle from the previous lesson, we’ve labeled the length \(b\) and the width \(h,\) so it’s area is \(bh.\)

Using the Properties of Triangles

The area of a rectangle is the base, \(b,\) times the height, \(h.\)

We can divide this rectangle into two congruent triangles (see the figure below). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or \(\frac{1}{2}bh.\) This example helps us see why the formula for the area of a triangle is \(A=\frac{1}{2}bh.\)

Using the Properties of Triangles

A rectangle can be divided into two triangles of equal area. The area of each triangle is one-half the area of the rectangle.

The formula for the area of a triangle is \(A=\frac{1}{2}bh,\) where \(b\) is the base and \(h\) is the height.

To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a \(\text{90°}\) angle with the base. The figure below shows three triangles with the base and height of each marked.

Using the Properties of Triangles

The height \(h\) of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a \(\text{90°}\) angle with the base.

Definition: Triangle Properties

For any triangle \(\text{Δ}ABC,\) the sum of the measures of the angles is \(\text{180°}.\)

\(m\text{∠}A+m\text{∠}B+m\text{∠}C=\text{180°}\)

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

\(P=a+b+c\)

The area of a triangle is one-half the base, \(b,\) times the height, \(h.\)

\(A=\frac{1}{2}\phantom{\rule{0.1em}{0ex}}bh\)

Using the Properties of Triangles

Optional Video: Area of a Triangle

Example

Find the area of a triangle whose base is \(11\) inches and whose height is \(8\) inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.Using the Properties of Triangles
Step 2. Identify what you are looking for.the area of the triangle
Step 3. Name. Choose a variable to represent it.let A = area of the triangle
Step 4.Translate.

 

Write the appropriate formula.

 

Substitute.

 

Using the Properties of Triangles

Step 5. Solve the equation.Using the Properties of Triangles
Step 6. Check:

 

Using the Properties of Triangles

 
Step 7. Answer the question.The area is 44 square inches.

Example

The perimeter of a triangular garden is \(24\) feet. The lengths of two sides are \(4\) feet and \(9\) feet. How long is the third side?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.Using the Properties of Triangles
Step 2. Identify what you are looking for.length of the third side of a triangle
Step 3. Name. Choose a variable to represent it.Let c = the third side
Step 4.Translate.

 

Write the appropriate formula.

 

Substitute in the given information.

 

Using the Properties of Triangles

Step 5. Solve the equation.Using the Properties of Triangles
Step 6. Check:

 

Using the Properties of Triangles

 
Step 7. Answer the question.The third side is 11 feet long.

Example

The area of a triangular church window is \(90\) square meters. The base of the window is \(15\) meters. What is the window’s height?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.Using the Properties of Triangles
Step 2. Identify what you are looking for.height of a triangle
Step 3. Name. Choose a variable to represent it.Let h = the height
Step 4.Translate.

 

Write the appropriate formula.

 

Substitute in the given information.

 

Using the Properties of Triangles

Step 5. Solve the equation.Using the Properties of Triangles
Step 6. Check:

 

Using the Properties of Triangles

 
Step 7. Answer the question.The height of the triangle is 12 meters.

Isosceles and Equilateral Triangles

Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle. The figure below shows both types of triangles.

Using the Properties of Triangles

In an isosceles triangle, two sides have the same length, and the third side is the base. In an equilateral triangle, all three sides have the same length.

Definition: Isosceles and Equilateral Triangles

An isosceles triangle has two sides the same length.

An equilateral triangle has three sides of equal length.

Example

The perimeter of an equilateral triangle is \(93\) inches. Find the length of each side.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.Using the Properties of Triangles

 

Perimeter = 93 in.

Step 2. Identify what you are looking for.length of the sides of an equilateral triangle
Step 3. Name. Choose a variable to represent it.Let s = length of each side
Step 4.Translate.

 

Write the appropriate formula.

 

Substitute.

 

Using the Properties of Triangles

Step 5. Solve the equation.Using the Properties of Triangles
Step 6. Check:

 

Using the Properties of Triangles

 

Using the Properties of Triangles

 
Step 7. Answer the question.Each side is 31 inches.

Example

Arianna has \(156\) inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of

\(60\) inches. How long can she make the two equal sides?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.Using the Properties of Triangles

 

P = 156 in.

Step 2. Identify what you are looking for.the lengths of the two equal sides
Step 3. Name. Choose a variable to represent it.Let s = the length of each side
Step 4.Translate.

 

Write the appropriate formula.

 

Substitute in the given information.

 

Using the Properties of Triangles

Step 5. Solve the equation.Using the Properties of Triangles
Step 6. Check:

 

Using the Properties of Triangles

 
Step 7. Answer the question.Arianna can make each of the two equal sides 48 inches long.

Optional Video: Area of a Triangle with Fractions

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