To find the length of the longer diagonal in a parallelogram where two adjacent sides are known, as well as the shorter diagonal, we can use the following formula for the diagonals $d_1$ and $d_2$ of a parallelogram:

$d_1^2 + d_2^2 = 2(a^2 + b^2)$

where:

• $a$ and $b$ are the lengths of the two adjacent sides of the parallelogram,
• $d_1$ and $d_2$ are the lengths of the diagonals.

Given:

• $a = 9$,
• $b = 15$,
• $d_1 = 14$ (the shorter diagonal).

Step-by-Step Solution

1. Substitute Known Values into the equation:

   $14^2 + d_2^2 = 2(9^2 + 15^2)$

2. Calculate $9^2$ and $15^2$:

   $9^2 = 81 \quad \text{and} \quad 15^2 = 225$

3. Sum $9^2$ and $15^2$:

   $81 + 225 = 306$

4. Calculate $2 \times 306$:

   $2 \times 306 = 612$

5. Substitute back into the equation:

   $196 + d_2^2 = 612$

6. Solve for $d_2^2$:

   $d_2^2 = 612 - 196 = 416$

7. Take the square root to find $d_2$:

   $d_2 = \sqrt{416} \approx 20.4$

Answer

The length of the longer diagonal is approximately 20.4 units.

Would you like more details or have any questions?

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Relative Questions:

1. How would the formula change if we only knew one side and one diagonal?
2. What properties distinguish diagonals in a parallelogram compared to those in a rectangle?
3. How do the side lengths affect the lengths of the diagonals in a parallelogram?
4. Can a parallelogram be uniquely determined by its diagonals alone?
5. How do we calculate the area of a parallelogram given its diagonals?

Tip: In any quadrilateral, the sum of the squares of the diagonals can often help relate side lengths, especially in parallelograms and rectangles.