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:

\[ \boxed{26.2} \]