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

Question

GPT-4o mini · Answer

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.

Length between (1,5) and (4,1):

$d_{1} = \sqrt{(4 - 1)^{2} + (1 - 5)^{2}} = \sqrt{(3)^{2} + (- 4)^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$

Length between (4,1) and (−4,−5):

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

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:

$Perimeter \approx 26.2$

Thus, the answer is 26.2.