`DeltaDEF `is right angled at `E. EC `is the altitude. CF is 18 cm and FD is 26 cm. What is the length of FE?

Options

A. `2sqrt(13)` cm
B. `6sqrt(13)` cm
C. 12 cm
D. 15 cm

Answer

To find the length of \( FE \) in triangle \( DEF \) which is right-angled at \( E \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- \( CF = 18 \) cm
- \( FD = 26 \) cm

2. **Calculate the Length of \( CD \)**:
- Since \( CF + FD = CD \), we can find \( CD \):
\[ CD = FD - CF = 26 \, \text{cm} - 18 \, \text{cm} = 8 \, \text{cm} \]

3. **Use the Altitude Formula**:
- The formula for the altitude \( EC \) from point \( E \) to hypotenuse \( CD \) in a right triangle is given by:
\[ EC^2 = DC \times CF \]
- Here, \( DC = 8 \) cm and \( CF = 18 \) cm. Substituting the values:
\[ EC^2 = 8 \times 18 = 144 \]
- Therefore, taking the square root:
\[ EC = \sqrt{144} = 12 \, \text{cm} \]

4. **Calculate the Length of \( EF \)**:
- In triangle \( CEF \), we can use the Pythagorean theorem:
\[ EF^2 = CF^2 + EC^2 \]
- Substituting the known values:
\[ EF^2 = 18^2 + 12^2 = 324 + 144 = 468 \]
- Taking the square root:
\[ EF = \sqrt{468} \]

5. **Simplify \( \sqrt{468} \)**:
- We can simplify \( \sqrt{468} \):
\[ \sqrt{468} = \sqrt{36 \times 13} = 6\sqrt{13} \]

### Final Answer:
Thus, the length of \( FE \) is \( 6\sqrt{13} \) cm.

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