Question

construction company is building the foundation for a new office building. The diagram below shows the lengths of some sides of the foundation.

A rectangle with length 80 feet and width 70 feet. Point A is on the top side and it is 40 feet away from the top left vertex and point B is on right side and it is 40 feet away from the bottom right vertex. A and B joined to form a right triangle at the top right corner.

What is the length of the side between point A and point B?

A.
30 ft

B.
40 ft

C.
50 ft

D.
70 ft

Answer 1

To find the length of the side between point A and point B, we can use the given coordinates of points A and B on the rectangle.

1. **Identify the coordinates of Point A and Point B:**
- The rectangle has length 80 feet and width 70 feet.
- The top left vertex of the rectangle can be labeled as (0, 70), the top right vertex (80, 70), etc.
- Point A is located 40 feet from the top left vertex along the top edge. This means point A's coordinates are (40, 70).
- Point B is located 40 feet from the bottom right vertex along the right edge. This means point B's coordinates are (80, 30).

2. **Calculate the distance between Points A and B:**
We will use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where \((x_1, y_1)\) are the coordinates of point A and \((x_2, y_2)\) are the coordinates of point B.

Substituting the coordinates:
- \(A(40, 70)\)
- \(B(80, 30)\)

\[
d = \sqrt{(80 - 40)^2 + (30 - 70)^2}
\]
\[
d = \sqrt{(40)^2 + (-40)^2}
\]
\[
d = \sqrt{1600 + 1600}
\]
\[
d = \sqrt{3200}
\]
\[
d = 40\sqrt{2}
\]

Now, we can approximate \(40\sqrt{2}\) since \(\sqrt{2} \approx 1.414\):
\[
d \approx 40 \times 1.414 \approx 56.56
\]

3. **Choose the closest answer:**
The calculated length of the side between points A and B is approximately \(56.56\) feet. In the list of options provided:
- A. 30 ft
- B. 40 ft
- C. 50 ft
- D. 70 ft

The closest option is **C. 50 ft**.