Question

Use the information to answer the question.

Information
A city has a garden in the shape of an equilateral triangle with 40 foot sides. There is a walkway that runs from a gate located at the midpoint of one side of the equilateral triangle to a fountain at the opposite vertex.





Question
Approximately how long is the walkway?

Answer options with 4 options
A.
23.09 feet

B.
28.28 feet

C.
34.64 feet

D.
40.00 feet

Answer 1

To find the length of the walkway from the midpoint of one side of an equilateral triangle to the opposite vertex, we can use some basic geometry.

Given:
- Each side of the equilateral triangle is 40 feet.
- Let’s label the triangle as ABC, where AB = AC = BC = 40 feet.
- The midpoint of side BC is M.

The coordinates of the vertices of the triangle can be placed in a 2D coordinate system for easier calculations:
- Point B (0, 0)
- Point C (40, 0)
- To find the coordinates of point A, we can calculate its position using the properties of an equilateral triangle:

The height (h) of the equilateral triangle can be calculated by:
\[
h = \frac{\sqrt{3}}{2} \times \text{side length} = \frac{\sqrt{3}}{2} \times 40 = 20\sqrt{3} \approx 34.64 \text{ feet}
\]

Thus, the coordinates for point A would be:
- Point A (20, \(20\sqrt{3}\)) - which is approximately (20, 34.64).

The midpoint, M, of side BC is:
- M (20, 0)

Now we need to find the distance from M (20, 0) to point A (20, \(20\sqrt{3}\)).

Using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates:
\[
d = \sqrt{(20 - 20)^2 + (20\sqrt{3} - 0)^2} = \sqrt{0 + (20\sqrt{3})^2} = \sqrt{400 \times 3} = \sqrt{1200} = 20\sqrt{3}
\]

Calculating \(20\sqrt{3}\) gives:
\[
20\sqrt{3} \approx 20 \times 1.732 \approx 34.64 \text{ feet}
\]

Thus, the length of the walkway is approximately 34.64 feet.

The correct answer is:
**C. 34.64 feet**