If a Quadrilateral having sides AB = 6 , BC = 7, CD = 5 . Find AD , if diagonals intersect at right angle

Answer

To find the length of side AD in the quadrilateral ABCD, where the diagonals intersect at right angles, we can use the property that states:

**If the diagonals of a quadrilateral intersect at right angles, then:**
\[ AB^2 + CD^2 = BC^2 + AD^2 \]

### Step-by-Step Solution:

1. **Identify the given values:**
- \( AB = 6 \)
- \( BC = 7 \)
- \( CD = 5 \)
- We need to find \( AD \).

2. **Write down the equation based on the property:**
\[ AB^2 + CD^2 = BC^2 + AD^2 \]

3. **Substitute the known values into the equation:**
\[ 6^2 + 5^2 = 7^2 + AD^2 \]

4. **Calculate the squares:**
\[ 36 + 25 = 49 + AD^2 \]

5. **Simplify the left side:**
\[ 61 = 49 + AD^2 \]

6. **Isolate \( AD^2 \):**
\[ AD^2 = 61 - 49 \]
\[ AD^2 = 12 \]

7. **Take the square root to find \( AD \):**
\[ AD = \sqrt{12} \]
\[ AD = 2\sqrt{3} \]

### Final Answer:
The length of side \( AD \) is \( 2\sqrt{3} \).

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