Two sides of a parallelogram are 9 and 15 units in length. The length of the shorter diagonal of the parallelogram is 14 units. Find the length of the long diagonal

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?

---

**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.

Two sides of a parallelogram are 9 and 15 units in length. The length of the shorter diagonal of the parallelogram is 14 units. Find the length of the long diagonal.

Explore how to calculate the length of the longer diagonal in a parallelogram with sides of 9 and 15 units and a shorter diagonal of 14 units. This problem uses key geometry formulas and is ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation