In `DeltaDEF` ,measure of angle E is `90^(@)`.If `secD=(25)/(7)andDE=1.4`cm ,the what is the length (in cm.)of side DF?

Options

A. 5
B. 4.8
C. 4
D. 5.6

Answer

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

### Step-by-Step Solution:

1. **Understanding Secant**:
The secant of an angle in a right triangle is defined as the ratio of the length of the hypotenuse to the length of the adjacent side. In this case, we have:
\[ \sec D = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{DF}{DE} \]
Given that \(\sec D = \frac{25}{7}\) and \(DE = 1.4 \, \text{cm}\).

2. **Setting Up the Equation**:
From the definition of secant, we can set up the equation:
\[ \frac{DF}{DE} = \frac{25}{7} \]
Substituting the value of \(DE\):
\[ \frac{DF}{1.4} = \frac{25}{7} \]

3. **Cross-Multiplying**:
To solve for \(DF\), we cross-multiply:
\[ DF \cdot 7 = 25 \cdot 1.4 \]

4. **Calculating the Right Side**:
Now we calculate \(25 \cdot 1.4\):
\[ 25 \cdot 1.4 = 35 \]

5. **Solving for DF**:
Now we have:
\[ 7DF = 35 \]
Dividing both sides by 7 gives:
\[ DF = \frac{35}{7} = 5 \, \text{cm} \]

### Final Answer:
The length of side DF is \(5 \, \text{cm}\).

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