Chapter 11.7, Problem 2PSA

a.

To determine

To calculate:Ratio of the areas of similar figures with sides $5$ and $2$

Answer to Problem 2PSA

The ratio of the areas of similar figures with sides $5$ and $2$ is $25:4$

Explanation of Solution

Given: The sides are $5$ and $2$

Concept used: Here, we are using Angle Angle similarity.

Calculation: Here, we know that both the triangles are similar by Angle Angle similarity.

So, Ratio of areas

  → ${\left(\frac{5}{2}\right)}^2=\frac{ar\left({\Delta }_1\right)}{ar\left({\Delta }_2\right)}$

  → $\frac{ar\left({\Delta }_1\right)}{ar\left({\Delta }_2\right)}=\frac{25}{4}$

Conclusion: Hence, the ratio of the areas of similar figures with sides $5$ and $2$ is $25:4$

b.

To determine

To calculate: Ratio of the areas of similar figures with sides $6$ and $9$

Answer to Problem 2PSA

The ratio of the areas of similar figures with sides $6$ and $9$ is $4:9$

Explanation of Solution

Given: The sides are $6$ and $9$

Calculation: Here, both rectangles are similar. So, all angles are right angles.

Ratio of areas are:

  ${\left(\frac{6}{9}\right)}^2=\frac{ar\left(rec\tan gl{e}_1\right)}{ar\left(rec\tan gl{e}_2\right)}$

  → $\frac{ar\left(rec\tan gl{e}_1\right)}{ar\left(rec\tan gl{e}_2\right)}=\frac{4}{9}$

Conclusion: Hence, the ratio of the areas of similar figures with sides $6$ and $9$ is $4:9$

c.

To determine

To calculate:Ratio of the areas of similar figures with sides $8$ and $4$

Answer to Problem 2PSA

The ratio of the areas of similar figures with sides $8$ and $4$ is $4:1$

Explanation of Solution

Given: The sides are $8$ and $4$

Concept used: Here, we are using the concept of Angle Angle similarity.

Calculation: Here, we know that both the figures are similar by Angle Angle similarity.

So, the ratio of areas:

  $\frac{ar\left({\Delta }_1\right)}{ar\left({\Delta }_2\right)}={\left(\frac{8}{4}\right)}^2$

  $\frac{ar\left({\Delta }_1\right)}{ar\left({\Delta }_2\right)}=\frac{4}{1}$

Conclusion: Hence, the ratio of the areas of similar figures with sides $8$ and $4$ is $4:1$

d.

To determine

To calculate: Ratio of the areas of similar figures with sides $2$ and $6$

Answer to Problem 2PSA

The ratio of the areas of similar figures with sides $2$ and $6$ is $1:9$

Explanation of Solution

Given: The sides are $2$ and $6$

Calculation: Here, we know that both the figures are similar by Angle Angle similarity.

So, the ratio of areas:

  $\frac{ar\left({\Delta }_1\right)}{ar\left({\Delta }_2\right)}={\left(\frac{2}{6}\right)}^2$

  $\frac{ar\left({\Delta }_1\right)}{ar\left({\Delta }_2\right)}=\frac{1}{9}$

Conclusion: Hence, the ratio of the areas of similar figures with sides $2$ and $6$ is $1:9$