The areas of two similar figures are in a ratio of 49 :16. What is the ratio of their sides?Question 7

1. Break Down the Problem

To find the ratio of the sides of two similar figures given the ratio of their areas, we can denote:

• Let the ratio of their areas be $A_1 : A_2 = 49 : 16$.
• We will determine the ratio of their corresponding sides.

2. Relevant Concepts

The areas of similar figures are proportional to the square of the ratio of their corresponding sides. Therefore, if $s_1$ and $s_2$ are the sides of the two figures, the relationship can be expressed as: $\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2$

3. Analysis and Detail

Given: $\frac{A_1}{A_2} = \frac{49}{16}$ We can set the ratio of the sides $\frac{s_1}{s_2}$ as $\frac{x}{y}$: $\frac{49}{16} = \left(\frac{x}{y}\right)^2$ To find $\frac{x}{y}$, we take the square root of both sides: $\frac{x}{y} = \sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}} = \frac{7}{4}$

4. Verify and Summarize

Thus, the ratio of the sides $s_1 : s_2$ is established as $7 : 4$.

Final Answer

The ratio of their sides is $7 : 4$.