In `DeltaDEF` measure of angle E .is `90^(@)` .If `cosD=(8)/(17),andDE=16cm`., then what is the length (in cm.)of side DF?

Options

A. 30
B. 20
C. 26
D. 34

Answer

To find the length of side DF in triangle DEF, where angle E is 90 degrees, we can use the information given about cos D and the length of side DE.

### Step-by-Step Solution:

1. **Identify the triangle and given values**:
- Triangle DEF is a right triangle with angle E = 90°.
- We are given that DE = 16 cm and cos D = 8/17.

2. **Understanding the cosine function**:
- In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse.
- For angle D, we have:
\[ \cos D = \frac{\text{Adjacent side (DE)}}{\text{Hypotenuse (DF)}} \]
- Therefore, we can write:
\[ \cos D = \frac{DE}{DF} = \frac{16}{DF} \]

3. **Set up the equation using the given cosine value**:
- We know that:
\[ \cos D = \frac{8}{17} \]
- Thus, we can set up the equation:
\[ \frac{16}{DF} = \frac{8}{17} \]

4. **Cross-multiply to solve for DF**:
- Cross-multiplying gives us:
\[ 16 \cdot 17 = 8 \cdot DF \]
- Simplifying this, we have:
\[ 272 = 8 \cdot DF \]

5. **Divide both sides by 8 to find DF**:
- To isolate DF, divide both sides by 8:
\[ DF = \frac{272}{8} = 34 \text{ cm} \]

### Final Answer:
The length of side DF is **34 cm**.