Solving Applications Using Properties of Triangles
Solving 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.
Solve Geometry Applications.
- Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
- Identify what we are looking for.
- Label what we are looking for by choosing a variable to represent it.
- Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Solve the equation using good algebra techniques.
- Check the answer by substituting it back into the equation solved in step 5 and by making sure it makes sense in the context of the problem.
- Answer the question with a complete sentence.
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 the figure below is called \(\text{△}ABC.\)
Triangle $ABC$ has vertices $A$, $B$, and $C$. The lengths of the sides are $a$, $b$, and $c$.
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 the figure below,
\(m\text{∠}A+m\text{∠}B+m\text{∠}C=180\text{°}\)
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.
\(P=a+b+c\)
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 the figure below.
The formula for the area of \(\text{△}ABC\) is \(A=\frac{1}{2}bh,\) where $b$ is the base and $h$ is the height.
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.
Example
The measures of two angles of a triangle are 55 and 82 degrees. Find the measure of the third angle.
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 measure of the third angle in a triangle |
| Step 3. Name. Choose a variable to represent it. | Let \(x=\) the measure of the angle. |
| Step 4. Translate. | |
| Write the appropriate formula and substitute. | \(m\angle A+m\angle B+m\angle C=180\) |
| Step 5. Solve the equation. | \(\begin{array}{ccc}\hfill 55+82+x& =\hfill & 180\hfill \\ \hfill 137+x& =\hfill & 180\hfill \\ \hfill x& =\hfill & 43\hfill \end{array}\) |
| Step 6. Check. \(\begin{array}{ccc}\hfill 55+82+43& \stackrel{?}{=}\hfill & 180\hfill \\ \hfill 180& =\hfill & 180✓\hfill \end{array}\) |
|
| Step 7. Answer the question. | The measure of the third angle is 43 degrees. |
Example
The perimeter of a triangular garden is 24 feet. The lengths of two sides are four feet and nine 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 and substitute. | |
| Substitute in the given information. | |
| Step 5. Solve the equation. | |
| Step 6. Check. \(\begin{array}{ccc}\hfill P& =\hfill & a+b+c\hfill \\ \hfill 24& \stackrel{?}{=}\hfill & 4+9+11\hfill \\ \hfill 24& =\hfill & 24✓\hfill \end{array}\) |
|
| 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. | Area \(=90{m}^{2}\) |
| 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. \(\begin{array}{ccc}\hfill A& =\hfill & \frac{1}{2}bh\hfill \\ \hfill 90& \stackrel{?}{=}\hfill & \frac{1}{2}\cdot 15\cdot 12\hfill \\ \hfill 90& =\hfill & 90✓\hfill \end{array}\) |
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| Step 7. Answer the question. | The height of the triangle is 12 meters. |
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 \(90\text{°}\) angle, which we usually mark with a small square in the corner.
Right Triangle
A right triangle has one \(90\text{°}\) angle, which is often marked with a square at the vertex.
Example
One angle of a right triangle measures \(28\text{°}.\) What is the measure of the third angle?
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 measure of an angle |
| Step 3. Name. Choose a variable to represent it. | Let \(x=\) the measure of an angle. |
| Step 4. Translate. | \(m\angle A+m\angle B+m\angle C=180\) |
| Write the appropriate formula and substitute. | \(x+90+28=180\) |
| Step 5. Solve the equation. | \(\begin{array}{ccc}\hfill x+118& =\hfill & 180\hfill \\ \hfill x& =\hfill & 62\hfill \end{array}\) |
| Step 6. Check. \(\begin{array}{ccc}\hfill 180& \stackrel{?}{=}\hfill & 90+28+62\hfill \\ \hfill 180& =\hfill & 180✓\hfill \end{array}\) |
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| Step 7. Answer the question. | The measure of the third angle is 62°. |
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.
Example
The measure of one angle of a right triangle is 20 degrees more than the measure of the smallest angle. Find the measures of all three angles.
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what you are looking for. | the measures of all three angles |
| Step 3. Name. Choose a variable to represent it. | \(\phantom{\rule{0.24em}{0ex}}\)Let \(a={1}^{\text{st}}\) angle. \(a+20={2}^{\text{nd}}\) angle \(\phantom{\rule{1.7em}{0ex}}90={3}^{\text{rd}}\) angle (the right angle) |
| Draw the figure and label it with the given information | |
| Step 4. Translate | |
| Write the appropriate formula. Substitute into the formula. |
|
| Step 5. Solve the equation. | 55 90 third angle |
| Step 6. Check. \(\begin{array}{ccc}\hfill 35+55+90& \stackrel{?}{=}\hfill & 180\hfill \\ \hfill 180& =\hfill & 180✓\hfill \end{array}\) |
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| Step 7. Answer the question. | The three angles measure 35°, 55°, and 90°. |
This lesson is part of:
Math Models and Geometry II