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Answer :
To find the perimeter of the triangle with vertices at [tex]\((1,5)\)[/tex], [tex]\((4,1)\)[/tex], and [tex]\((-4,-5)\)[/tex], we need to calculate the distance between each pair of points, which will give us the lengths of the sides of the triangle. Then, we add these lengths together to find the perimeter. Here's how you calculate each side:
1. Calculate the distance between [tex]\((1,5)\)[/tex] and [tex]\((4,1)\)[/tex]:
- Use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- Plug in the coordinates:
[tex]\[
\text{Distance} = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
2. Calculate the distance between [tex]\((4,1)\)[/tex] and [tex]\((-4,-5)\)[/tex]:
- Use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\][/tex]
3. Calculate the distance between [tex]\((-4,-5)\)[/tex] and [tex]\((1,5)\)[/tex]:
- Use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125} \approx 11.2
\][/tex]
4. Add up the lengths of the sides to find the perimeter:
[tex]\[
\text{Perimeter} = 5 + 10 + 11.2 = 26.2
\][/tex]
Therefore, the perimeter of the triangle, rounded to the nearest tenth of a unit, is [tex]\(26.2\)[/tex].
1. Calculate the distance between [tex]\((1,5)\)[/tex] and [tex]\((4,1)\)[/tex]:
- Use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- Plug in the coordinates:
[tex]\[
\text{Distance} = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
2. Calculate the distance between [tex]\((4,1)\)[/tex] and [tex]\((-4,-5)\)[/tex]:
- Use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\][/tex]
3. Calculate the distance between [tex]\((-4,-5)\)[/tex] and [tex]\((1,5)\)[/tex]:
- Use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125} \approx 11.2
\][/tex]
4. Add up the lengths of the sides to find the perimeter:
[tex]\[
\text{Perimeter} = 5 + 10 + 11.2 = 26.2
\][/tex]
Therefore, the perimeter of the triangle, rounded to the nearest tenth of a unit, is [tex]\(26.2\)[/tex].
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