Chapter 7.5, Problem 10WE

To determine

To Evaluate: The scale factor of the similarity of the given triangle.

Answer to Problem 10WE

The scalar factor of the similarity of the given triangles is $\frac{4}{5}$ .

Explanation of Solution

Given information:

AB	BC	AC	TR	RI	TI
6	8	10	10	7.5	12.5

According to AA similarity theorem;

  $\begin{array}{cc}& \angle A\cong \angle D\\ & \angle B\cong \angle E\\ & \text{then,}\\ & \text{Δ}ABC\sim \text{Δ}DEF\end{array}$

According SSS similarity theorem ;

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

Then,

  $\text{Δ}ABC\sim \text{Δ}DEF$

According to SAS similarity theorem;

  $\begin{array}{l}\angle A\cong \angle D\hfill \\ \frac{AB}{DE}=\frac{AC}{DF}\hfill \end{array}$

Then,

  $\text{Δ}ABC\sim \text{Δ}DEF$

Scalar factor : If the two polygons are similar, then the ratio of the length of two corresponding sides is called the scalar factor of the similarity.

Now consider one triangle has vertices A,B, and C and another triangle has vertices T, R, and I.

AB	BC	AC	TR	RI	TI
6	8	10	10	7.5	12.5

Now,

Compare the longest sides $\frac{AC}{TI}=\frac{10}{12.5}=\frac{4}{5}$

Compare the shortest side $\frac{AB}{RI}=\frac{6}{7.5}=\frac{4}{5}$

Compare the remaining sides

  $\frac{BC}{TR}=\frac{8}{10}=\frac{4}{5}$

Therefore,

  $\begin{array}{cc}& \frac{AC}{TI}=\frac{AB}{RI}=\frac{BC}{TR}=\frac{4}{5}\\ & \text{Thus, the scalar factor is }\frac{4}{5}\end{array}$

Hence, the given triangles are similar and the scalar factor of the similarity of the given triangles is $\frac{4}{5}$ .