We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Guides d'étude > Prealgebra

Using the Properties of Triangles to Solve Problems

Learning Outcomes

  • Find the area, base and height of a triangle
  • Find the length of one side of a triangle given the perimeter and two other lengths
 

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 below, we’ve labeled the length bb and the width hh, so it’s area is bhbh.

The area of a rectangle is the base, bb, times the height, hh. A rectangle is shown. The side is labeled h and the bottom is labeled b. The center says A equals bh. We can divide this rectangle into two congruent triangles (see the image 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 12bh\frac{1}{2}bh. This example helps us see why the formula for the area of a triangle is A=12bhA=\frac{1}{2}bh. A rectangle can be divided into two triangles of equal area. The area of each triangle is one-half the area of the rectangle. A rectangle is shown. A diagonal line is drawn from the upper left corner to the bottom right corner. The side of the rectangle is labeled h and the bottom is labeled b. Each triangle says one-half bh. To the right of the rectangle, it says The formula for the area of a triangle is A=12bhA=\frac{1}{2}bh, where bb is the base and hh 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 90∘\text{90}^ \circ angle with the base. The image below shows three triangles with the base and height of each marked. The height hh of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a 90∘\text{90}^ \circ angle with the base. Three triangles are shown. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.

Triangle Properties

For any triangle ΔABC\Delta ABC, the sum of the measures of the angles is 180∘\text{180}^ \circ. m∠A+m∠B+m∠C=180∘m\angle{A}+m\angle{B}+m\angle{C}=180^\circ The perimeter of a triangle is the sum of the lengths of the sides. P=a+b+cP=a+b+c The area of a triangle is one-half the base, bb, times the height, hh. A=12bhA=\frac{1}{2}bh A triangle is shown. The vertices are labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.
 

example

Find the area of a triangle whose base is 1111 inches and whose height is 88 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. A=44A=44 square inches.
Step 6. Check: A=12bhA=\frac{1}{2}bh 44=?12(11)844\stackrel{?}{=}\frac{1}{2}(11)8 44=44✓44=44\checkmark
Step 7. Answer the question. The area is 4444 square inches.
 

try it

[ohm_question]146525[/ohm_question]
 

example

The perimeter of a triangular garden is 2424 feet. The lengths of two sides are 44 feet and 99 feet. How long is the third side?

Answer: 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. 24=13+c24=13+c 11=c11=c
Step 6. Check: P=a+b+cP=a+b+c 24=?4+9+1124\stackrel{?}{=}4+9+11 24=24✓24=24\checkmark
Step 7. Answer the question. The third side is 1111 feet long.

 

try it

[ohm_question]146526[/ohm_question]
   

example

The area of a triangular church window is 9090 square meters. The base of the window is 1515 meters. What is the window’s height?

Answer: 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. 90=152h90=\frac{15}{2}h 12=h12=h
Step 6. Check: A=12bhA=\frac{1}{2}bh 90=?12⋅15⋅1290\stackrel{?}{=}\frac{1}{2}\cdot 15\cdot 12 90=90✓90=90\checkmark
Step 7. Answer the question. The height of the triangle is 1212 meters.

 

try it

[ohm_question]146528[/ohm_question] [ohm_question]146527[/ohm_question]
 

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 image below shows both types 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. Two triangles are shown. All three sides of the triangle on the left are labeled s. It is labeled

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 9393 inches. Find the length of each side.

Answer: Solution

Step 1. Read the problem. Draw the figure and label it with the given information. . Perimeter = 9393 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. 93=3s93=3s 31=s31=s
Step 6. Check: . 93=?31+31+3193\stackrel{?}{=}31+31+31 93=93✓93=93\checkmark
Step 7. Answer the question. Each side is 3131 inches.

 

try it

[ohm_question]146529[/ohm_question]
 

example

Arianna has 156156 inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of 6060 inches. How long can she make the two equal sides?

Answer: Solution

Step 1. Read the problem. Draw the figure and label it with the given information. . P = 156156 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. 156=2s=60156=2s=60 96=2s96=2s 48=s48=s
Step 6. Check: p=a+b+cp=a+b+c 156=?48+60+48156\stackrel{?}{=}48+60+48 156=156✓156=156\checkmark
Step 7. Answer the question. Arianna can make each of the two equal sides 4848 inches long.

 

try it

[ohm_question]146531[/ohm_question]
 

Licenses & Attributions