Questions: The areas of two similar squares are 16 m^2 and 49 m^2. What is the scale factor of their side lengths?

Transcript text: The areas of two similar squares are $16 \mathrm{~m}^{2}$ and $49 \mathrm{~m}^{2}$. What is the scale factor of their side lengths?

Solution

Solution Steps

Step 1: Understand the Problem

We are given the areas of two similar squares and need to find the scale factor of their side lengths. The areas of the squares are $16 \, \text{m}^2$ and $49 \, \text{m}^2$.

Step 2: Recall the Relationship Between Areas and Side Lengths

For similar figures, the ratio of their areas is the square of the ratio of their corresponding side lengths. If the side lengths of the squares are $a$ and $b$, then: \[ \left(\frac{a}{b}\right)^2 = \frac{\text{Area of first square}}{\text{Area of second square}} \]

Step 3: Set Up the Equation

Let $a$ be the side length of the smaller square and $b$ be the side length of the larger square. The given areas are $16 \, \text{m}^2$ and $49 \, \text{m}^2$, so: \[ \left(\frac{a}{b}\right)^2 = \frac{16}{49} \]

Step 4: Solve for the Scale Factor

To find the scale factor $\frac{a}{b}$, we take the square root of both sides of the equation: \[ \frac{a}{b} = \sqrt{\frac{16}{49}} = \frac{\sqrt{16}}{\sqrt{49}} = \frac{4}{7} \]