Applying the Dilation Factor

Oct 13, 2025 by miodeya.com 67 views

Introduction

In this problem, we are given a triangle that has been dilated by a scale factor of 2. This means that all the sides of the triangle have been multiplied by a factor of 2. We are also given that $\cos a^{\circ} = \frac{3}{5}$ and that $\overline{FD}$ measures 6 units. Our goal is to find the length of $\overline{DE}$ .

Understanding the Problem

To solve this problem, we need to understand the concept of dilation and how it affects the sides of a triangle. When a triangle is dilated by a scale factor of 2, all the sides of the triangle are multiplied by a factor of 2. This means that if we have a triangle with sides of length 3, 4, and 5, after dilation, the new sides will have lengths of 6, 8, and 10.

Using the Cosine Function

We are given that $\cos a^{\circ} = \frac{3}{5}$ . This is a trigonometric function that relates the length of the adjacent side to the length of the hypotenuse in a right triangle. In this case, we can use the cosine function to find the length of $\overline{DE}$ .

Applying the Dilation Factor

Since the triangle has been dilated by a scale factor of 2, we need to multiply the length of $\overline{DE}$ by 2 to find the new length. This means that if $\overline{DE}$ measures 3.6 units, after dilation, it will measure 7.2 units.

Finding the Length of $\overline{DE}$

To find the length of $\overline{DE}$ , we can use the cosine function and the fact that the triangle has been dilated by a scale factor of 2. We know that $\cos a^{\circ} = \frac{3}{5}$ , and we can use this to find the length of $\overline{DE}$ .

Step 1: Draw a Diagram

Let's draw a diagram of the triangle and label the sides. We can see that $\overline{FD}$ measures 6 units, and we want to find the length of $\overline{DE}$ .

Step 2: Use the Cosine Function

We can use the cosine function to find the length of $\overline{DE}$ . Since $\cos a^{\circ} = \frac{3}{5}$ , we can set up the following equation:

\cos a^{\circ} = \frac{\overline{DE}}{\overline{FD}}

Step 3: Substitute the Values

We can substitute the values of $\cos a^{\circ}$ and $\overline{FD}$ into the equation:

\frac{3}{5} = \frac{\overline{DE}}{6}

Step 4: Solve for $\overline{DE}$

We can solve for $\overline{DE}$ by multiplying both sides of the equation by 6:

\overline{DE} = \frac{3}{5} \times 6

Step 5: Simplify the Expression

We can simplify the expression by multiplying the fractions:

\overline{DE} = \frac{18}{5}

Step 6: Apply the Dilation Factor

Since the triangle has been dilated by a scale factor of 2, we need to multiply the length of $\overline{DE}$ by 2 to find the new length:

\overline{DE} = 2 \times \frac{18}{5}

Step 7: Simplify the Expression

We can simplify the expression by multiplying the fractions:

\overline{DE} = \frac{36}{5}

Step 8: Convert the Fraction to a Decimal

We can convert the fraction to a decimal by dividing the numerator by the denominator:

\overline{DE} = 7.2

Conclusion

In this problem, we were given a triangle that had been dilated by a scale factor of 2. We were also given that $\cos a^{\circ} = \frac{3}{5}$ and that $\overline{FD}$ measures 6 units. Our goal was to find the length of $\overline{DE}$ . We used the cosine function and the fact that the triangle had been dilated by a scale factor of 2 to find the length of $\overline{DE}$ . The final answer is $\boxed{7.2}$ units.

Final Answer

The final answer is $\boxed{7.2}$ units.