given trapezoid abcd, ab // dc, and ad perpendicular to ac. if ad=6, ab=bc=7, find bd

We are given a trapezoid \(ABCD\), where:
- \(AB \parallel DC\),
- \(AD \perp AC\),
- \(AD = 6\),
- \(AB = BC = 7\).

We are tasked with finding the length of diagonal \(BD\).

### Step-by-Step Solution:

1. **Analyzing the Geometry**:
   - The trapezoid has one pair of parallel sides (\(AB \parallel DC\)) and a right angle at \(D\) since \(AD \perp AC\).
   - Because \(AB = BC\), triangle \(ABC\) is isosceles with \(AB = BC\).

2. **Use the Pythagorean Theorem**:
   To find the length of \(BD\), we note that triangle \(ABD\) is a right triangle with the right angle at \(A\) because \(AD \perp AC\). We can use the Pythagorean theorem in triangle \(ABD\).

   - In triangle \(ABD\):
     \[
     BD^2 = AB^2 + AD^2
     \]

3. **Substitute the Given Values**:
   - \(AB = 7\),
   - \(AD = 6\).

   Using these values:
   \[
   BD^2 = 7^2 + 6^2 = 49 + 36 = 85
   \]

4. **Solve for \(BD\)**:
   \[
   BD = \sqrt{85}
   \]
   Approximating:
   \[
   BD \approx 9.22
   \]

Thus, the length of diagonal \(BD\) is approximately \(9.22\).

### Do you want further details or have any questions?

### Related Questions:
1. How would the solution change if \(AB \neq BC\)?
2. Can the Pythagorean theorem be applied to any triangle?
3. What is the significance of the right angle in solving for diagonal lengths?
4. How would you find the height of the trapezoid given different side lengths?
5. How can you find the area of the trapezoid with this information?

### Tip:
Always check for perpendicular or parallel lines in geometric shapes; they can simplify your calculations with right triangles or symmetry!

Given trapezoid ABCD, AB // DC, and AD perpendicular to AC. If AD = 6, AB = BC = 7, find BD.

Explore how to find the diagonal BD in a trapezoid ABCD where AB // DC, AD ⊥ AC, with AD = 6 and AB = BC = 7. Using the Pythagorean theorem, this solution is suitable for students in Grades 8-10 learning about right triangles and trapezoids.

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