Using the Properties of Triangles
Contents
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 \(bh.\)
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 \(A=\frac{1}{2}bh.\)
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.
Definition: Triangle Properties
For any triangle \(\text{Δ}ABC,\) the sum of the measures of the angles is \(\text{180°}.\)
The perimeter of a triangle is the sum of the lengths of the sides.
The area of a triangle is one-half the base, \(b,\) times the height, \(h.\)
\(A=\frac{1}{2}\phantom{\rule{0.1em}{0ex}}bh\)
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. | |
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. | |
Step 5. Solve the equation. | |
Step 6. Check: | |
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. | |
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. | |
Step 5. Solve the equation. | |
Step 6. Check: | |
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. | |
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. | |
Step 5. Solve the equation. | |
Step 6. Check: | |
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.
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. | 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. | |
Step 5. Solve the equation. | |
Step 6. Check: | |
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. | 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. | |
Step 5. Solve the equation. | |
Step 6. Check: | |
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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