Explore a wide range of topics and get answers from experts on IDNLearn.com. Our platform provides prompt, accurate answers from experts ready to assist you with any question you may have.

Find the perimeter of the triangle whose vertices are [tex](-2, -5), (3, -5)[/tex], and [tex](3, 7)[/tex]. Write the exact answer. Do not round.

Answer:


Sagot :

To find the perimeter of the triangle with vertices [tex]\((-2, -5)\)[/tex], [tex]\((3, -5)\)[/tex], and [tex]\((3, 7)\)[/tex], we can follow these steps:

1. Determine the lengths of the sides of the triangle using the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:

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

2. Calculate the length of the side between [tex]\((-2, -5)\)[/tex] and [tex]\((3, -5)\)[/tex]. These points have the same [tex]\(y\)[/tex]-coordinate, so the distance is:

[tex]\[ \text{side 1} = \sqrt{(3 - (-2))^2 + (-5 - (-5))^2} = \sqrt{(3 + 2)^2 + 0^2} = \sqrt{5^2} = 5 \][/tex]

3. Calculate the length of the side between [tex]\((3, -5)\)[/tex] and [tex]\((3, 7)\)[/tex]. These points have the same [tex]\(x\)[/tex]-coordinate, so the distance is:

[tex]\[ \text{side 2} = \sqrt{(3 - 3)^2 + (7 - (-5))^2} = \sqrt{0 + (7 + 5)^2} = \sqrt{12^2} = 12 \][/tex]

4. Calculate the length of the side between [tex]\((-2, -5)\)[/tex] and [tex]\((3, 7)\)[/tex]. The distance is:

[tex]\[ \text{side 3} = \sqrt{(3 - (-2))^2 + (7 - (-5))^2} = \sqrt{(3 + 2)^2 + (7 + 5)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \][/tex]

5. Add the lengths of the three sides to find the perimeter:

[tex]\[ \text{Perimeter} = \text{side 1} + \text{side 2} + \text{side 3} = 5 + 12 + 13 = 30 \][/tex]

Therefore, the perimeter of the triangle is [tex]\(30\)[/tex].