Understanding the Relationship Between Sine and Scale Factor

Oct 13, 2025 by windailys.com 70 views

A Triangle Was Dilated by a Scale Factor of 6: Understanding the Relationship Between Sine and Scale Factor

When dealing with dilations in geometry, it's essential to understand how the scale factor affects the measurements of the sides of a triangle. In this problem, we're given a triangle that has been dilated by a scale factor of 6, and we need to find the length of one of its sides, $\overline{EF}$ , given that $\sin a ^{\circ} = \frac{4}{5}$ and $\overline{DE}$ measures 30 units.

Understanding the Relationship Between Sine and Scale Factor

The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, we're given that $\sin a ^{\circ} = \frac{4}{5}$ . This means that the ratio of the length of the side opposite angle $a$ to the length of the hypotenuse is $\frac{4}{5}$ .

When a triangle is dilated by a scale factor of 6, all of its sides are multiplied by a factor of 6. This means that the length of the side opposite angle $a$ is multiplied by 6, and the length of the hypotenuse is also multiplied by 6.

Applying the Scale Factor to the Given Information

We're given that $\overline{DE}$ measures 30 units. Since the triangle has been dilated by a scale factor of 6, the length of $\overline{DE}$ is multiplied by 6. Therefore, the length of $\overline{DE}$ after dilation is $6 \times 30 = 180$ units.

Using the Sine Ratio to Find the Length of $\overline{EF}$

We know that $\sin a ^{\circ} = \frac{4}{5}$ , and we've found that the length of $\overline{DE}$ after dilation is 180 units. We can use the sine ratio to find the length of $\overline{EF}$ .

Let's denote the length of $\overline{EF}$ as $x$ . Since $\overline{EF}$ is opposite angle $a$ , we can set up the following equation:

\frac{x}{180} = \frac{4}{5}

To solve for $x$ , we can cross-multiply:

5x = 4 \times 180

5x = 720

Dividing both sides by 5, we get:

x = \frac{720}{5}

x = 144

However, this is not the correct answer. We need to consider the scale factor of 6.

Considering the Scale Factor of 6

Since the triangle has been dilated by a scale factor of 6, the length of $\overline{EF}$ is multiplied by 6. Therefore, the length of $\overline{EF}$ after dilation is $6 \times 24 = 144$ units.

However, we need to find the length of $\overline{EF}$ before dilation. To do this, we can divide the length of $\overline{EF}$ after dilation by the scale factor of 6:

\frac{144}{6} = 24

Therefore, the length of $\overline{EF}$ before dilation is 24 units.

Conclusion

In this problem, we were given a triangle that has been dilated by a scale factor of 6, and we needed to find the length of one of its sides, $\overline{EF}$ , given that $\sin a ^{\circ} = \frac{4}{5}$ and $\overline{DE}$ measures 30 units. We used the sine ratio to find the length of $\overline{EF}$ before dilation, and we found that it is 24 units.

Final Answer

The final answer is $\boxed{24}$ units.