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7 of 77 of 7 Items 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?(2 points) Responses 36.9 36.9 26.2 26.2 32.8 32.8 Skip to navigation page 1 of 1

Answer 1

To find the perimeter of the triangle formed by the points (1,5), (4,1), and (−4,−5), we first need to calculate the lengths of each side using the distance formula:

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

**Step 1: Calculate the lengths of the sides.**

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)**:

\[
d_3 = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \approx 11.18
\]

**Step 2: Calculate the perimeter.**

Now we can find the perimeter \(P\) by adding the lengths of the sides:

\[
P = d_1 + d_2 + d_3 = 5 + 10 + 5\sqrt{5}
\]

Using \(\sqrt{5} \approx 2.236\):

\[
5\sqrt{5} \approx 5 \times 2.236 \approx 11.18
\]

So,

\[
P \approx 5 + 10 + 11.18 \approx 26.18
\]

Rounding to the nearest tenth, the perimeter of the triangle is approximately:

\[
\text{Perimeter} \approx 26.2
\]

Thus, the answer is **26.2**.