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

10 units
10 units

21.2 units
21.2 units

26.2 units
26.2 units

11.2 units

Answer 1

To find the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we need to calculate the lengths of the three 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:

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

Let's calculate each side:

1. **Length between \((1, 5)\) and \((4, 1)\)**:

\[
d_1 = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

2. **Length between \((4, 1)\) and \((-4, -5)\)**:

\[
d_2 = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]

3. **Length between \((-4, -5)\) and \((1, 5)\)**:

\[
d_3 = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \approx 11.2
\]

Now we can find the perimeter of the triangle by adding the lengths of the three sides:

\[
\text{Perimeter} = d_1 + d_2 + d_3 = 5 + 10 + 11.2 \approx 26.2
\]

Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is:

\(\boxed{26.2}\) units.