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? Show all steps that Nathan must take to find the solution.

Answer 1

To find the perimeter of the triangle with vertices at the points \((1,5)\), \((4,1)\), and \((-4,-5)\), we will follow these steps:

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

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

We will calculate the distance between each pair of vertices.

2. **Calculate the length of side AB \( (1,5) \) to \( (4,1) \)**:

\[
AB = \sqrt{(4 - 1)^2 + (1 - 5)^2}
\]
\[
= \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]

3. **Calculate the length of side BC \( (4,1) \) to \( (-4,-5) \)**:

\[
BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2}
\]
\[
= \sqrt{(-8)^2 + (-6)^2}
\]
\[
= \sqrt{64 + 36}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]

4. **Calculate the length of side CA \( (1,5) \) to \( (-4,-5) \)**:

\[
CA = \sqrt{(-4 - 1)^2 + (-5 - 5)^2}
\]
\[
= \sqrt{(-5)^2 + (-10)^2}
\]
\[
= \sqrt{25 + 100}
\]
\[
= \sqrt{125}
\]
\[
= 5\sqrt{5} \approx 11.2 \text{ (rounding to the nearest tenth)}
\]

5. **Calculate the perimeter of the triangle** by adding the lengths of the sides:

\[
\text{Perimeter} = AB + BC + CA
\]
\[
= 5 + 10 + 5\sqrt{5}
\]
\[
\approx 5 + 10 + 11.2 \quad \text{(using } 5\sqrt{5} \approx 11.2\text{)}
\]
\[
= 26.2
\]

6. **Round the perimeter** to the nearest tenth:

The perimeter of the triangle is approximately **26.2** units.

Thus, the final answer is:

\[
\boxed{26.2}
\]