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? (1 point)
Responses

26.2 units
26.2 units

10 units
10 units

21.2 units
21.2 units

11.2 units

Answer 1

To find the perimeter of the triangle with vertices at the points \(A(1, 5)\), \(B(4, 1)\), and \(C(-4, -5)\), we first need to calculate the lengths of each side of the triangle using the distance formula:

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

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

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

3. **Length \(CA\)**:
\[
CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} \approx 11.2
\]

Now, we can calculate the perimeter of the triangle by adding the lengths of all sides:

\[
\text{Perimeter} = AB + BC + CA = 5 + 10 + 11.2 = 26.2
\]

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