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# Unknown Dimensions of Triangles

## Use the formula A = (bh)/2 to solve for the unknown variable

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Unknown Dimensions of Triangles

Bill and Takeen are running backs for the high school football team. They need to be in a split running back formation, which is a triangular pattern, and stay on a line 5 yards behind the quarterback. The area of the triangle should be approximately 20 square yards. How far apart should Bill and Takeen stand from one another?

In this concept, you will learn how to find the unknown dimensions in a triangle.

### Finding Unknown Dimensions in a Triangle

The area of a triangle can be found by multiplying the base times the height and then dividing the product by 2.

\begin{align*}A = \frac{bh}{2}\end{align*}

This equation can also be used if you are given the area and asked to find either the base or the height.

Here’s an example:

A triangle has an area of \begin{align*}44 \ m^2\end{align*}. The base of the triangle is 8 m. What is its height?

First, write the formula and substitute the values that are given.

\begin{align*}\begin{array}{rcl} A &=& \frac{bh}{2} \\ 44 &=& \frac{8h}{2} \end{array}\end{align*}

Next, begin isolating your unknown by multiplying both sides of the equation by 2.

\begin{align*}\begin{array}{rcl} 44 \times 2 &=& \frac{8h}{2} \times 2 \\ 88 &=& 8h \end{array}\end{align*}

Then, divide both sides by 8.

\begin{align*}\begin{array}{rcl} 88 &=& 8h\\ 11 &=& h \end{array}\end{align*}

The answer is 11 m. The height of the triangle is 11 m.

You can check your answer by placing all the dimensions back into the equation.

\begin{align*}\begin{array}{rcl} A &=& \frac{bh}{2} \\ 44 &=& \frac{(8)(11)}{2} \\ 44 &=& \frac{88}{2} \\ 44 &=& 44 \end{array}\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about the running backs, Bill and Takeen.

They needed to know how far apart they should stand from one another to maintain a 20 square foot triangular pattern with the quarterback who is on a line 5 yards in front of them.

First, draw a picture and label it.

Next, fill in the formula with the values that are given.

\begin{align*}\begin{array}{rcl} A &=& \frac{bh}{2} \\ 20 &=& \frac{(b)(5)}{2} \end{array}\end{align*}

Begin isolating your unknown by multiplying both sides of the equation by 2.

\begin{align*}\begin{array}{rcl} 20 \times 2 &=& \frac{(b)(5)}{2} \times 2 \\ 40 &=& 10b \end{array}\end{align*}

Then, divide both sides by 10

\begin{align*}4=b\end{align*}

The answer is 4 yards. Bill and Taken should stand about 4 yards apart from one another.

#### Example 2

A triangle has a \begin{align*}\text{base} = 4 \ \text{inches}\end{align*} and an \begin{align*}\text{area} = 6 \ \text{sq. inches}\end{align*}. What is its height?

First, write the formula and substitute the values that are given.

\begin{align*}\begin{array}{rcl} A &=& \frac{bh}{2} \\ 6 &=& \frac{4h}{2} \end{array}\end{align*}

Next, begin isolating your unknown by multiplying both sides of the equation by 2.

\begin{align*}\begin{array}{rcl} 6 \times 2 &=& \frac{4h}{2} \times 2 \\ 12 &=& 4h \end{array}\end{align*}

Then, divide both sides by 4.

\begin{align*}3=h\end{align*}

The answer is 3 inches. The height of the triangle is 3 inches.

#### Example 3

A triangle’s \begin{align*}\text{height} = 3 \ \text{feet}\end{align*}. Its \begin{align*}\text{area} = 7.5 \ \text{sq. feet}\end{align*}. What is the triangle’s base?

First, write the formula and substitute the values that are given.

\begin{align*}\begin{array}{rcl} A &=& \frac{bh}{2} \\ 7.5 &=& \frac{(b)(3)}{2} \end{array}\end{align*}

Next, begin isolating your unknown by multiplying both sides of the equation by 2.

\begin{align*}\begin{array}{rcl} 7.5 \times 2 &=& \frac{(b)(3)}{2} \times 2 \\ 15 &=& 3b \end{array}\end{align*}

Then, divide both sides by 3.

\begin{align*}5 = b\end{align*}

The answer is 5 feet. The base of the triangle is 5 feet.

#### Example 4

A triangle’s \begin{align*}\text{base} = 7 \ \text{meters}\end{align*}, \begin{align*}\text{area} = 17.5 \ sq.\end{align*} meters, what is the height?

First, write the formula and substitute the values that are given.

\begin{align*}\begin{array}{rcl} A &=& \frac{bh}{2} \\ 6 &=& \frac{4h}{2} \end{array}\end{align*}

Next, begin isolating your unknown by multiplying both sides of the equation by 2.

\begin{align*}\begin{array}{rcl} 6 \times 2 &=& \frac{4h}{2} \times 2 \\ 12 &=& 4h \end{array}\end{align*}

Then, divide both sides by 4.

\begin{align*}3=h\end{align*}

The answer is 3 inches. The height of the triangle is 3 inches.

#### Example 5

A triangle has an area of 11.25 square feet and a base length of 4.5 ft. What is the height of the triangle?
First, write the formula and substitute the values that are given.

\begin{align*}\begin{array}{rcl} A &=& \frac{bh}{2} \\ 11.25 &=& \frac{4.5h}{2} \end{array}\end{align*}

Next, begin isolating your unknown by multiplying both sides of the equation by 2.

\begin{align*}\begin{array}{rcl} 11.25 \times 2 &=& \frac{4.5h}{2} \times 2 \\ 22.5 &=& 4.5h \end{array}\end{align*}

Then, divide both sides by 4.5

\begin{align*}5=h\end{align*}

The answer is 5 feet. The height of the triangle is 5 feet.

### Review

Find the missing base or height given the area and one other dimension.

1. Area = 13.5 sq. meters, Base = 9 meters
2. Area = 21 sq. meters, Base = 7 meters
3. Area = 12 sq. meters, Base = 8 meters
4. Area = 33 sq. ft, Base = 11 feet
5. Area = 37.5 sq. ft. Base = 15 feet
6. Area = 60 sq. ft., height = 10 ft.
7. Area = 20.25 sq. in, height = 4.5 in
8. Area = 72 sq. in, height = 8 in
9. Area = 22.5 sq. feet, height = 5 feet
10. Area = 12 sq. in, height = 4 in
11. Area = 45 sq. in, height = 9 in
12. Area = 84 sq. ft, height = 12 ft
13. Area = 144 sq. in, height = 16 in
14. Area = 144.5 sq. ft, height = 17 ft.
15. Area = 123.5 sq. in, height = 19 in

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### Vocabulary Language: English

TermDefinition
Area Area is the space within the perimeter of a two-dimensional figure.
Base The side of a triangle parallel with the bottom edge of the paper or screen is commonly called the base. The base of an isosceles triangle is the non-congruent side in the triangle.
Height The height of a triangle is the perpendicular distance from the base of the triangle to the opposite vertex of the triangle.
Triangle A triangle is a polygon with three sides and three angles.