Question

Find the missing measures for the pair of similar triangles if $\Delta ABC~\Delta DEF$.

$a=3,b=4,c=5,d=4.5$

Short answer

The missing measures for the pairs of similar triangles are$e=6$and $f=7.5$.

Step-by-step solution

Step 1. Write the definition of similar triangles.

The two triangles are said to be similar if the measures of the corresponding angles are equal and if the two triangles are similar then the measures of their corresponding sides are proportional.

Step 2. Draw the rough sketch of the triangles ΔABC and ΔDEF.

The rough sketch of the triangles$\Delta ABC$ and$\Delta DEF$ are:

Step 3. Find the missing measures for the pair of similar triangles if ΔABC~ΔDEF.

As, the triangles$\Delta ABC$ and$\Delta DEF$ are similar, therefore the measures of their corresponding sides are proportional.

That implies,

$\frac{BC}{EF}=\frac{AC}{DF}=\frac{AB}{DE}$

Therefore, it can be said that:

$$\begin{gathered} \frac{BC}{EF}=\frac{AC}{DF} \\ \frac{a}{d}=\frac{b}{e} \\ \frac{3}{4.5}=\frac{4}{e} \\ 3e=18 \\ e=\frac{18}{3} \\ e=6 \end{gathered}$$

$$\begin{gathered} \frac{AC}{DF}=\frac{AB}{DE} \\ \frac{b}{e}=\frac{c}{f} \\ \frac{4}{6}=\frac{5}{f} \\ 4f=30 \\ f=\frac{30}{4} \\ f=7.5 \end{gathered}$$

Therefore, the missing measures for the pairs of similar triangles are$e=6$ and $f=7.5$.