Question
A triangle has a perimeter of exactly 24 units. Which of the following could be the vertices of the triangle?
(1 point)
Responses
Left-parenthesis negative 1 comma 3 right-parenthesis, left-parenthesis 2 comma negative 1 right-parenthesis, left-parenthesis negative 1 comma negative 1 right-parenthesis
Image with alt text: Left-parenthesis negative 1 comma 3 right-parenthesis, left-parenthesis 2 comma negative 1 right-parenthesis, left-parenthesis negative 1 comma negative 1 right-parenthesis
Left-parenthesis 6 comma 0 right-parenthesis, left-parenthesis 6 comma 7 right-parenthesis, left-parenthesis 0 comma 7 right-parenthesis
Image with alt text: Left-parenthesis 6 comma 0 right-parenthesis, left-parenthesis 6 comma 7 right-parenthesis, left-parenthesis 0 comma 7 right-parenthesis
Left-parenthesis negative 1 comma negative 1 right-parenthesis, left-parenthesis negative 6 comma negative 13 right-parenthesis, left-parenthesis negative 9 comma negative 9 right-parenthesis
Image with alt text: Left-parenthesis negative 1 comma negative 1 right-parenthesis, left-parenthesis negative 6 comma negative 13 right-parenthesis, left-parenthesis negative 9 comma negative 9 right-parenthesis
Left-parenthesis negative 3 comma negative 4 right-parenthesis, left-parenthesis 3 comma negative 4 right-parenthesis, left-parenthesis 3 comma 4 right-parenthesis
(1 point)
Responses
Left-parenthesis negative 1 comma 3 right-parenthesis, left-parenthesis 2 comma negative 1 right-parenthesis, left-parenthesis negative 1 comma negative 1 right-parenthesis
Image with alt text: Left-parenthesis negative 1 comma 3 right-parenthesis, left-parenthesis 2 comma negative 1 right-parenthesis, left-parenthesis negative 1 comma negative 1 right-parenthesis
Left-parenthesis 6 comma 0 right-parenthesis, left-parenthesis 6 comma 7 right-parenthesis, left-parenthesis 0 comma 7 right-parenthesis
Image with alt text: Left-parenthesis 6 comma 0 right-parenthesis, left-parenthesis 6 comma 7 right-parenthesis, left-parenthesis 0 comma 7 right-parenthesis
Left-parenthesis negative 1 comma negative 1 right-parenthesis, left-parenthesis negative 6 comma negative 13 right-parenthesis, left-parenthesis negative 9 comma negative 9 right-parenthesis
Image with alt text: Left-parenthesis negative 1 comma negative 1 right-parenthesis, left-parenthesis negative 6 comma negative 13 right-parenthesis, left-parenthesis negative 9 comma negative 9 right-parenthesis
Left-parenthesis negative 3 comma negative 4 right-parenthesis, left-parenthesis 3 comma negative 4 right-parenthesis, left-parenthesis 3 comma 4 right-parenthesis
Answers
Answer
The correct answer is (-3, -4), (3, -4), (3,4).
Answered by
GPT 3.5
To determine if a triangle can have a perimeter of 24 units, we need to check if the sum of any two sides is greater than the third side.
Let's calculate the distances between each pair of points and check if they satisfy the triangle inequality.
1. For the first set of points, we have:
Distance between (-1, 3) and (2, -1) = √[(2-(-1))^2 + (-1-3)^2] = √[3^2 + (-4)^2] = 5
Distance between (-1, 3) and (-1, -1) = √[(-1-(-1))^2 + (-1-3)^2] = √[0^2 + (-4)^2] = 4
Distance between (2, -1) and (-1, -1) = √[(2-(-1))^2 + (-1-(-1))^2] = √[3^2 + 0^2] = 3
Sum of any two sides:
5 + 4 = 9 (which is smaller than 3)
5 + 3 = 8 (which is smaller than 4)
4 + 3 = 7 (which is smaller than 5)
Therefore, the first set of points cannot form a triangle with a perimeter of 24 units.
2. For the second set of points, we have:
Distance between (6, 0) and (6, 7) = √[0^2 + (7-0)^2] = √[7^2] = 7
Distance between (6, 0) and (0, 7) = √[(6-0)^2 + (7-0)^2] = √[6^2 + 7^2] = √[36 + 49] = √85 ≈ 9.22
Distance between (6, 7) and (0, 7) = √[(6-0)^2 + (7-7)^2] = √[6^2 + 0^2] = √36 = 6
Sum of any two sides:
7 + 9.22 ≈ 16.22 (which is smaller than 6)
7 + 6 = 13 (which is smaller than 9.22)
9.22 + 6 ≈ 15.22 (which is smaller than 7)
Therefore, the second set of points cannot form a triangle with a perimeter of 24 units.
3. For the third set of points, we have:
Distance between (-1, -1) and (-6, -13) = √[(-6-(-1))^2 + (-13-(-1))^2] = √[(-5)^2 + (-12)^2] = √[25 + 144] = √169 = 13
Distance between (-1, -1) and (-9, -9) = √[(-1-(-9))^2 + (-1-(-9))^2] = √[8^2 + 8^2] = √[64 + 64] = √128 ≈ 11.31
Distance between (-6, -13) and (-9, -9) = √[(-6-(-9))^2 + (-13-(-9))^2] = √[3^2 + (-4)^2] = √[9 + 16] = √25 = 5
Sum of any two sides:
13 + 11.31 ≈ 24.31 (which is greater than 5)
13 + 5 = 18 (which is greater than 11.31)
11.31 + 5 ≈ 16.31 (which is greater than 13)
Therefore, the third set of points can form a triangle with a perimeter of 24 units.
4. For the fourth set of points, we have:
Distance between (-3, -4) and (3, -4) = √[(3-(-3))^2 + (-4-(-4))^2] = √[6^2 + 0^2] = √36 = 6
Distance between (3, -4) and (3, 4) = √[0^2 + (4-(-4))^2] = √[0^2 + 8^2] = √[64] = 8
Distance between (-3, -4) and (3, 4) = √[(3-(-3))^2 + (4-(-4))^2] = √[6^2 + 8^2] = √[36 + 64] = √100 = 10
Sum of any two sides:
6 + 8 = 14 (which is smaller than 10)
6 + 10 = 16 (which is smaller than 8)
8 + 10 = 18 (which is greater than 6)
Therefore, the fourth set of points cannot form a triangle with a perimeter of 24 units.
Based on this analysis, the only set of points that could form a triangle with a perimeter of exactly 24 units is:
(-1, -1), (-6, -13), and (-9, -9).
Let's calculate the distances between each pair of points and check if they satisfy the triangle inequality.
1. For the first set of points, we have:
Distance between (-1, 3) and (2, -1) = √[(2-(-1))^2 + (-1-3)^2] = √[3^2 + (-4)^2] = 5
Distance between (-1, 3) and (-1, -1) = √[(-1-(-1))^2 + (-1-3)^2] = √[0^2 + (-4)^2] = 4
Distance between (2, -1) and (-1, -1) = √[(2-(-1))^2 + (-1-(-1))^2] = √[3^2 + 0^2] = 3
Sum of any two sides:
5 + 4 = 9 (which is smaller than 3)
5 + 3 = 8 (which is smaller than 4)
4 + 3 = 7 (which is smaller than 5)
Therefore, the first set of points cannot form a triangle with a perimeter of 24 units.
2. For the second set of points, we have:
Distance between (6, 0) and (6, 7) = √[0^2 + (7-0)^2] = √[7^2] = 7
Distance between (6, 0) and (0, 7) = √[(6-0)^2 + (7-0)^2] = √[6^2 + 7^2] = √[36 + 49] = √85 ≈ 9.22
Distance between (6, 7) and (0, 7) = √[(6-0)^2 + (7-7)^2] = √[6^2 + 0^2] = √36 = 6
Sum of any two sides:
7 + 9.22 ≈ 16.22 (which is smaller than 6)
7 + 6 = 13 (which is smaller than 9.22)
9.22 + 6 ≈ 15.22 (which is smaller than 7)
Therefore, the second set of points cannot form a triangle with a perimeter of 24 units.
3. For the third set of points, we have:
Distance between (-1, -1) and (-6, -13) = √[(-6-(-1))^2 + (-13-(-1))^2] = √[(-5)^2 + (-12)^2] = √[25 + 144] = √169 = 13
Distance between (-1, -1) and (-9, -9) = √[(-1-(-9))^2 + (-1-(-9))^2] = √[8^2 + 8^2] = √[64 + 64] = √128 ≈ 11.31
Distance between (-6, -13) and (-9, -9) = √[(-6-(-9))^2 + (-13-(-9))^2] = √[3^2 + (-4)^2] = √[9 + 16] = √25 = 5
Sum of any two sides:
13 + 11.31 ≈ 24.31 (which is greater than 5)
13 + 5 = 18 (which is greater than 11.31)
11.31 + 5 ≈ 16.31 (which is greater than 13)
Therefore, the third set of points can form a triangle with a perimeter of 24 units.
4. For the fourth set of points, we have:
Distance between (-3, -4) and (3, -4) = √[(3-(-3))^2 + (-4-(-4))^2] = √[6^2 + 0^2] = √36 = 6
Distance between (3, -4) and (3, 4) = √[0^2 + (4-(-4))^2] = √[0^2 + 8^2] = √[64] = 8
Distance between (-3, -4) and (3, 4) = √[(3-(-3))^2 + (4-(-4))^2] = √[6^2 + 8^2] = √[36 + 64] = √100 = 10
Sum of any two sides:
6 + 8 = 14 (which is smaller than 10)
6 + 10 = 16 (which is smaller than 8)
8 + 10 = 18 (which is greater than 6)
Therefore, the fourth set of points cannot form a triangle with a perimeter of 24 units.
Based on this analysis, the only set of points that could form a triangle with a perimeter of exactly 24 units is:
(-1, -1), (-6, -13), and (-9, -9).
Answered by
GPT 3.5
Apologies for the mistake. Let's re-calculate using the correct points.
For the corrected set of points, we have:
Distance between (-3, -4) and (3, -4) = √[(3-(-3))^2 + (-4-(-4))^2] = √[6^2 + 0^2] = √36 = 6
Distance between (3, -4) and (3, 4) = √[0^2 + (4-(-4))^2] = √[0^2 + 8^2] = √[64] = 8
Distance between (-3, -4) and (3, 4) = √[(3-(-3))^2 + (4-(-4))^2] = √[6^2 + 8^2] = √[36 + 64] = √100 = 10
Sum of any two sides:
6 + 8 = 14 (which is smaller than 10)
6 + 10 = 16 (which is smaller than 8)
8 + 10 = 18 (which is smaller than 6)
Therefore, the corrected set of points (-3, -4), (3, -4), and (3, 4) can form a triangle with a perimeter of exactly 24 units. Thank you for pointing out the error.
For the corrected set of points, we have:
Distance between (-3, -4) and (3, -4) = √[(3-(-3))^2 + (-4-(-4))^2] = √[6^2 + 0^2] = √36 = 6
Distance between (3, -4) and (3, 4) = √[0^2 + (4-(-4))^2] = √[0^2 + 8^2] = √[64] = 8
Distance between (-3, -4) and (3, 4) = √[(3-(-3))^2 + (4-(-4))^2] = √[6^2 + 8^2] = √[36 + 64] = √100 = 10
Sum of any two sides:
6 + 8 = 14 (which is smaller than 10)
6 + 10 = 16 (which is smaller than 8)
8 + 10 = 18 (which is smaller than 6)
Therefore, the corrected set of points (-3, -4), (3, -4), and (3, 4) can form a triangle with a perimeter of exactly 24 units. Thank you for pointing out the error.