Solve Proportion and Similar Figure Applications: example 8.78

$Δ A B C$ is similar to $Δ X Y Z$ . The lengths of two sides of each triangle are given. Find the lengths of the third sides.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

$Δ A B C$ is similar to $Δ X Y Z$ . The lengths of two sides of each triangle are given in the figure.

Find the length of side $a$ .

How did it go?

Step 1. Read the problem. Draw the figure and label it with the given information.	Figure is given.
Step 2. Identify what we are looking for.	the length of the sides of similar triangles
Step 3. Name the variables.	Let $a =$ length of the third side of $Δ A B C .$ $y =$ length of the third side of $Δ X Y Z$
Step 4. Translate.	Since the triangles are similar, the corresponding sides are proportional.
We need to write an equation that compares the side we are looking for to a known ratio. Since the side AB = 4 corresponds to the side XY = 3 we know $\frac{A B}{X Y} = \frac{4}{3}$ . So we write equations with $\frac{A B}{X Y}$ to find the sides we are looking for. Be careful to match up corresponding sides correctly.	$\frac{A B}{X Y} = \frac{B C}{Y Z} = \frac{A C}{X Z} .$
Substitute.
Step 5. Solve the equation.

Step 6. Check. $\begin{matrix} \begin{matrix} \frac{4}{3} & \overset{?}{=} & \frac{6}{4.5} \\ 4 (4.5) & \overset{?}{=} & 6 (3) \\ 18 & = & 18 ✓ \end{matrix} & \begin{matrix} \frac{4}{3} & \overset{?}{=} & \frac{3.2}{2.4} \\ 4 (2.4) & \overset{?}{=} & 3.2 (3) \\ 9.6 & = & 9.6 ✓ \end{matrix} \end{matrix}$
Step 7. Answer the question.	The third side of $Δ A B C$ is 6 and the third side of $Δ X Y Z$ is 2.4.

Step 1. Read the problem. Draw the figure and label it with the given information.	Figure is given.
Step 2. Identify what we are looking for.	the length of the sides of similar triangles
Step 3. Name the variables.	Let $a =$ length of the third side of $Δ A B C .$ $y =$ length of the third side of $Δ X Y Z$
Step 4. Translate.	Since the triangles are similar, the corresponding sides are proportional.
We need to write an equation that compares the side we are looking for to a known ratio. Since the side AB = 4 corresponds to the side XY = 3 we know $\frac{A B}{X Y} = \frac{4}{3}$ . So we write equations with $\frac{A B}{X Y}$ to find the sides we are looking for. Be careful to match up corresponding sides correctly.	$\frac{A B}{X Y} = \frac{B C}{Y Z} = \frac{A C}{X Z} .$
Substitute.
Step 5. Solve the equation.

Step 6. Check. $\begin{matrix} \begin{matrix} \frac{4}{3} & \overset{?}{=} & \frac{6}{4.5} \\ 4 (4.5) & \overset{?}{=} & 6 (3) \\ 18 & = & 18 ✓ \end{matrix} & \begin{matrix} \frac{4}{3} & \overset{?}{=} & \frac{3.2}{2.4} \\ 4 (2.4) & \overset{?}{=} & 3.2 (3) \\ 9.6 & = & 9.6 ✓ \end{matrix} \end{matrix}$
Step 7. Answer the question.	The third side of $Δ A B C$ is 6 and the third side of $Δ X Y Z$ is 2.4.