Question

A triangle was dilated by a scale factor of 2. If cos a° = three fifths and segment FD measures 6 units, how long is segment DE?

triangle DEF in which angle F is a right angle, angle D measures a degrees, and angle E measures b degrees

segment DE = 3.6 units
segment DE = 8 units
segment DE = 10 units
segment DE = 12.4 units

Answer 1

To solve for segment DE in triangle DEF, we note a few important properties of the triangle.

1. **Understanding the angles**: Since angle F is a right angle and \( \angle D = a^\circ \), we can express the cosine of angle D (a) using the adjacent and hypotenuse relationship in triangle DEF:
\[
\cos a = \frac{\text{adjacent (side DF)}}{\text{hypotenuse (side DE)}}
\]

2. Given that \( \cos a = \frac{3}{5} \), we will set:
- Side DF as the side adjacent to angle D
- Side DE as the hypotenuse

Therefore:
\[
\frac{DF}{DE} = \frac{3}{5}
\]

3. **Finding side DE**:
Using the information provided:
- Segment FD measures 6 units, which we assume corresponds to the adjacent side DF in this context.
- Since we need to find DE (the hypotenuse):
\[
DF = 6
\]

Substitute DF and solve for DE:
\[
\frac{6}{DE} = \frac{3}{5}
\]

Cross-multiply and solve for DE:
\[
3 \cdot DE = 6 \cdot 5
\]
\[
3 \cdot DE = 30
\]
\[
DE = \frac{30}{3} = 10 \text{ units}
\]

4. **Factor of dilation**:
The problem states that the triangle was dilated by a scale factor of 2. Therefore, we need to multiply our previously calculated DE by this scale factor:
\[
DE' = 10 \times 2 = 20 \text{ units}
\]

However, the problem asks us about possible values of segment DE in the undeformed triangle:

5. The options provided are:
- 3.6 units
- 8 units
- 10 units
- 12.4 units

Since we calculated that segment DE in the original triangle measures **10 units**, that corresponds with the option listed.

Thus, the answer is:

**Segment DE = 10 units.**