In `triangleDEF` measure of angle E is 90°. If sinD = 15/17, and DE = 4cm, then what is the length (in cm) of side EF?

Options

A. 8.5
B. 7.5
C. 5
D. 6

Answer

To solve the problem step by step, we will use the properties of right triangles and the sine function.

### Step 1: Understand the triangle and given values
We have a right triangle DEF where angle E is 90°. We are given:
- \( \sin D = \frac{15}{17} \)
- \( DE = 4 \, \text{cm} \)

### Step 2: Identify the sides in relation to angle D
In triangle DEF:
- The side opposite angle D (which is DE) is the perpendicular side.
- The hypotenuse (DF) is opposite the right angle (E).
- The base (EF) is adjacent to angle D.

From the sine definition:
\[ \sin D = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{DE}{DF} \]
Substituting the known values:
\[ \frac{15}{17} = \frac{4}{DF} \]

### Step 3: Solve for DF (the hypotenuse)
Cross-multiplying gives:
\[ 15 \cdot DF = 4 \cdot 17 \]
\[ 15 \cdot DF = 68 \]
Now, divide both sides by 15:
\[ DF = \frac{68}{15} \approx 4.53 \, \text{cm} \]

### Step 4: Use the Pythagorean theorem to find EF
According to the Pythagorean theorem:
\[ DF^2 = DE^2 + EF^2 \]
Substituting the known values:
\[ \left(\frac{68}{15}\right)^2 = 4^2 + EF^2 \]
Calculating \( \left(\frac{68}{15}\right)^2 \):
\[ \frac{4624}{225} = 16 + EF^2 \]
Now, convert 16 to a fraction:
\[ 16 = \frac{360}{225} \]
So we have:
\[ \frac{4624}{225} = \frac{360}{225} + EF^2 \]
Subtract \( \frac{360}{225} \) from both sides:
\[ EF^2 = \frac{4624 - 360}{225} = \frac{4264}{225} \]

### Step 5: Calculate EF
Now take the square root to find EF:
\[ EF = \sqrt{\frac{4264}{225}} = \frac{\sqrt{4264}}{15} \]
Calculating \( \sqrt{4264} \):
\[ EF \approx \frac{65.4}{15} \approx 4.36 \, \text{cm} \]

### Final Answer
The length of side EF is approximately \( 4.36 \, \text{cm} \).