Question

Darnell is constructing a rectangle window frame. He measured the length, the width, and the diagonal as 26 inches, 32 inches, and √1700 inches.

Use the drop-down menus to answer the following questions of Darnell’s window frame.

What is the longest side?

What is the square of the longest side?

What is the sum of the squares of the two shorter sides?

Does the window frame form right triangles?

Answer 1

To determine the answers, let's analyze the dimensions provided.

1. **Longest Side**:
- The given dimensions are:
- Length = 26 inches
- Width = 32 inches
- Diagonal = \( \sqrt{1700} \) inches
- To find the longest side, we compare the three values:
- 26 inches
- 32 inches
- \( \sqrt{1700} \)

First, let's calculate \( \sqrt{1700} \):
\[
\sqrt{1700} = \sqrt{100 \times 17} = 10\sqrt{17} \approx 10 \times 4.123 = 41.23 \text{ inches}
\]
Thus, the diagonal \( \sqrt{1700} \) is approximately 41.23 inches, which is greater than both 26 inches and 32 inches.

**Answer**: The longest side is \( \sqrt{1700} \) inches.

2. **Square of the Longest Side**:
- The square of the longest side, \( \sqrt{1700} \), is:
\[
(\sqrt{1700})^2 = 1700
\]

**Answer**: The square of the longest side is 1700.

3. **Sum of the Squares of the Two Shorter Sides**:
- The lengths of the two shorter sides are 26 inches and 32 inches.
- Their squares:
- \( 26^2 = 676 \)
- \( 32^2 = 1024 \)
- Now adding these squares together:
\[
676 + 1024 = 1700
\]

**Answer**: The sum of the squares of the two shorter sides is 1700.

4. **Does the Window Frame Form Right Triangles?**:
- According to the Pythagorean theorem, for a right triangle with sides \( a \) and \( b \), and hypotenuse \( c \):
\[
c^2 = a^2 + b^2
\]
- In our case, we found that:
- \( c^2 = 1700 \) (the square of the diagonal)
- The sum \( a^2 + b^2 = 1700 \) (the sum of the squares of the shorter sides)
- Since \( c^2 = a^2 + b^2 \), it confirms that the triangle formed is indeed a right triangle.

**Answer**: Yes, the window frame forms right triangles.