7.02 Problem solving with triangles

We have learned that the area of a triangle is found by halving the product of its base and height. If we already know the area, along with one of the dimensions, we can use this relationship to find the remaining dimension.

The triangle below has an area of 20\text{ mm}^2, and a height of 5\text{ mm}. How can we determine the base of the triangle?

From the formula, we know that \text{Area}=\dfrac12\times\text{base}\times\text{height}, which means that 20=\dfrac12\times\text{base}\times\text{5}.

Let's set up the formula and solve for the missing variable.

We've found the base of the triangle is 8. But what should our units be?

We started with the given area in \text{mm}^2, which was found by multiplying the base in \text{mm} times the height in \text{mm}, or \text{mm} \times \text{mm}. We used the formula to work back and find the dimension of the base, which is a measure of length in \text{mm}. Therefore our base is 8 \text { mm} .

Examples

Example 1

Find the value of h if the area of this triangle is 48\text{ m}^2.

Worked Solution

Create a strategy

Use the area of a triangle formula and solve for h.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times b\times h	Use the area of a triangle formula
\displaystyle 48	\displaystyle =	\displaystyle \dfrac12\times4\times h	Substitute A=48 and b=4
\displaystyle 48	\displaystyle =	\displaystyle 2\times h	Evaluate the multiplication
\displaystyle \dfrac{48}2	\displaystyle =	\displaystyle \dfrac{2\times h}2	Divide both sides by 2
\displaystyle 24	\displaystyle =	\displaystyle h	Evaluate
\displaystyle h	\displaystyle =	\displaystyle 24\text{ m}	Solve for h

Watch question walkthrough

Example 2

Jacob has a triangular piece of fabric with an area of 50 \text{ cm}^2. The length of the base of the fabric is 10 \text { cm}, find the \text {height} of the fabric in meters.

Worked Solution

Create a strategy

Use the area of a triangle formula and substitute the values that we know. Solve for the missing dimension and convert the unit to meters.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times b\times h	Use the area of triangle formula
\displaystyle 50	\displaystyle =	\displaystyle \dfrac12\times10\times h	Substitute A=50 and b=10
\displaystyle 50	\displaystyle =	\displaystyle 5\times h	Evaluate the multiplication
\displaystyle \dfrac{50}5	\displaystyle =	\displaystyle \dfrac{5\times h}5	Divide both sides by 5
\displaystyle h	\displaystyle =	\displaystyle 10\text{ cm}	Evaluate
\displaystyle h	\displaystyle =	\displaystyle ⬚\text{ m}	Consider how many centimeters are in 1 meter
\displaystyle h	\displaystyle =	\displaystyle 10\text{ cm}\times\dfrac{1 \text{ m}}{100\text{ cm}}	Convert length
\displaystyle h	\displaystyle =	\displaystyle 0.10\text{ m}	Evaluate