Question

Nathan draws a triangle on the coordinate grid. He marks his points at (1, 5), (4, 1), and (-4, —5). What is the perimeter of the triangle, rounded to the nearest tenth of a unit? Explain the steps that Nathan must take to find the solution.

Answer 1

To calculate the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we'll follow these steps:

1. **Calculate the lengths of each side of the triangle** using the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

2. **Calculate the lengths of the three sides of the triangle:**

- **Side 1**: Between \((1, 5)\) and \((4, 1)\):
\[
d_1 = \sqrt{(4 - 1)^2 + (1 - 5)^2}
\]
\[
= \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

- **Side 2**: Between \((4, 1)\) and \((-4, -5)\):
\[
d_2 = \sqrt{(-4 - 4)^2 + (-5 - 1)^2}
\]
\[
= \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]

- **Side 3**: Between \((-4, -5)\) and \((1, 5)\):
\[
d_3 = \sqrt{(1 - (-4))^2 + (5 - (-5))^2}
\]
\[
= \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5}
\]
\[
\approx 5 \times 2.236 \approx 11.180 (for calculation purposes, you can round at the end as needed)
\]

3. **Sum the lengths of the sides** to find the perimeter:
\[
P = d_1 + d_2 + d_3 = 5 + 10 + 5\sqrt{5}
\]
Approximating \(5\sqrt{5}\), where \(\sqrt{5} \approx 2.236\):
\[
5\sqrt{5} \approx 5 \times 2.236 \approx 11.180
\]
Thus,
\[
P \approx 5 + 10 + 11.180 = 26.180
\]

4. **Round the perimeter to the nearest tenth**:
\[
P \approx 26.2
\]

So, the perimeter of the triangle rounded to the nearest tenth is approximately **26.2 units**.