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 find the perimeter of the triangle formed by the points (1,5), (4,1), and (-4,-5), Nathan can follow these steps:

1. **Label the points**:
- Let A = (1, 5)
- Let B = (4, 1)
- Let C = (-4, -5)

2. **Calculate the lengths of the sides using the distance formula**:
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

- **Length of side AB**:
\[
d_{AB} = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

- **Length of side BC**:
\[
d_{BC} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]

- **Length of side CA**:
\[
d_{CA} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5}
\]

3. **Calculate \(5\sqrt{5}\) numerically**:
Using an approximate value for \(\sqrt{5} \approx 2.236\):
\[
5\sqrt{5} \approx 5 \times 2.236 \approx 11.18
\]

4. **Calculate the perimeter**:
The perimeter \(P\) of the triangle is the sum of the lengths of the sides:
\[
P = d_{AB} + d_{BC} + d_{CA} = 5 + 10 + 5\sqrt{5} \approx 5 + 10 + 11.18 = 26.18
\]

5. **Round to the nearest tenth**:
Rounding \(26.18\) to the nearest tenth gives \(26.2\).

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