Chapter 7.5, Problem 8WE

a.

To determine

To find if the given triangles are similar and to find the scale factor if similar.

Answer to Problem 8WE

  $\Delta ABC\sim \Delta TRI$

Explanation of Solution

Given:

Two triangles $\Delta ABC$ and $\Delta TRI$ .

$AB$	$BC$	$AC$	$TR$	$RI$	$TI$
$6$	$8$	$10$	$9$	$12$	$15$

Calculation:

From the given section,

$AB$	$BC$	$AC$	$TR$	$RI$	$TI$
$6$	$8$	$10$	$9$	$12$	$15$

Drawing the figure,

  

Now to be a similar triangle,

  $\Rightarrow \frac{AB}{TR}=\frac{BC}{RI}=\frac{AC}{TI}$ to be satisfy then it can be written as $\angle A=\angle T,\angle B=\angle R\text{ and }\angle C=\angle I$

Then only $\Delta ABC\sim \Delta TRI$

Now putting the values from the table to check,

  $\Rightarrow \frac{AB}{TR}=\frac{6}{9}=\frac{2}{3}$

  $\Rightarrow \frac{BC}{RI}=\frac{8}{12}=\frac{2}{3}$

  $\Rightarrow \frac{AC}{TI}=\frac{10}{15}=\frac{2}{3}$

Then $\Rightarrow \frac{AB}{TR}=\frac{BC}{RI}=\frac{AC}{TI}$ it can also be concluded that $\angle A=\angle T,\angle B=\angle R\text{ and }\angle C=\angle I$

Conclusion:

Therefore, $\Delta ABC\sim \Delta TRI$ for the above given values or condition.

b.

To determine

To find if the given triangles are similar and to find the scale factor if similar.

Answer to Problem 8WE

  $\Delta ABC\text{ and }\Delta TRI$ are not similar

Explanation of Solution

Given:

Two triangles $\Delta ABC$ and $\Delta TRI$ .

$AB$	$BC$	$AC$	$TR$	$RI$	$TI$
$6$	$8$	$10$	$12$	$22$	$16$

Calculation:

From the given section,

$AB$	$BC$	$AC$	$TR$	$RI$	$TI$
$6$	$8$	$10$	$12$	$22$	$16$

Drawing the figure,

  

Now to be a similar triangle,

  $\Rightarrow \frac{AB}{TR}=\frac{BC}{RI}=\frac{AC}{TI}$ to be satisfy then it can be written as $\angle A=\angle T,\angle B=\angle R\text{ and }\angle C=\angle I$

Then only $\Delta ABC\sim \Delta TRI$

Now putting the values from the table to check,

  $\Rightarrow \frac{AB}{TR}=\frac{6}{12}=\frac{2}{4}$

  $\Rightarrow \frac{BC}{RI}=\frac{8}{22}=\frac{4}{11}$

  $\Rightarrow \frac{AC}{TI}=\frac{10}{16}=\frac{5}{8}$

Then $\Rightarrow \frac{AB}{TR}\ne \frac{BC}{RI}\ne \frac{AC}{TI}$ it can also be concluded that $\angle A\ne \angle T,\angle B\ne \angle R\text{ and }\angle C\ne \angle I$

Conclusion:

Therefore, $\Delta ABC\text{ and }\Delta TRI$ are not similar for the above given values or condition.

c.

To determine

To find if the given triangles are similar and to find the scale factor if similar.

Answer to Problem 8WE

  $\Delta ABC\text{ and }\Delta TRI$ are not similar

Explanation of Solution

Given:

Two triangles $\Delta ABC$ and $\Delta TRI$ .

$AB$	$BC$	$AC$	$TR$	$RI$	$TI$
$6$	$8$	$10$	$20$	$25$	$15$

Calculation:

From the given section,

$AB$	$BC$	$AC$	$TR$	$RI$	$TI$
$6$	$8$	$10$	$20$	$25$	$15$

Drawing the figure,

  

Now to be a similar triangle,

  $\Rightarrow \frac{AB}{TR}=\frac{BC}{RI}=\frac{AC}{TI}$ to be satisfy then it can be written as $\angle A=\angle T,\angle B=\angle R\text{ and }\angle C=\angle I$

Then only $\Delta ABC\sim \Delta TRI$

Now putting the values from the table to check,

  $\Rightarrow \frac{AB}{TR}=\frac{6}{20}=\frac{3}{10}$

  $\Rightarrow \frac{BC}{RI}=\frac{8}{25}$

  $\Rightarrow \frac{AC}{TI}=\frac{10}{15}=\frac{2}{3}$

Then $\Rightarrow \frac{AB}{TR}\ne \frac{BC}{RI}\ne \frac{AC}{TI}$ it can also be concluded that $\angle A\ne \angle T,\angle B\ne \angle R\text{ and }\angle C\ne \angle I$

Conclusion:

Therefore, $\Delta ABC\text{ and }\Delta TRI$ are not similar for the above given values or condition.

d.

To determine

To find if the given triangles are similar and to find the scale factor if similar.

Answer to Problem 8WE

  $\Delta ABC\text{ and }\Delta TRI$ are not similar

Explanation of Solution

Given:

Two triangles $\Delta ABC$ and $\Delta TRI$ .

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

Calculation:

From the given section,

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

Drawing the figure,

  

Now to be a similar triangle,

  $\Rightarrow \frac{AB}{TR}=\frac{BC}{RI}=\frac{AC}{TI}$ to be satisfy then it can be written as $\angle A=\angle T,\angle B=\angle R\text{ and }\angle C=\angle I$

Then only $\Delta ABC\sim \Delta TRI$

Now putting the values from the table to check,

  $\Rightarrow \frac{AB}{TR}=\frac{6}{10}=\frac{3}{5}$

  $\Rightarrow \frac{BC}{RI}=\frac{8}{7.5}$

  $\Rightarrow \frac{AC}{TI}=\frac{10}{12.5}$

Then $\Rightarrow \frac{AB}{TR}\ne \frac{BC}{RI}\ne \frac{AC}{TI}$ it can also be concluded that $\angle A\ne \angle T,\angle B\ne \angle R\text{ and }\angle C\ne \angle I$

Conclusion:

Therefore, $\Delta ABC\text{ and }\Delta TRI$ are not similar for the above given values or condition.