A triangle has a perimeter of exactly 24 units. Which of the following could be the vertices of the triangle?
A.) (-1,3),(2,-1),(-1,-1)
B.) (6,0),(6,7)(0,7)
C.) (-1,-1),(-6,-13),(-9,-9)
D.) (-3,-4),(3,-4),(3,4)
I am not sure on this one any help would be great
D - thank you so much - that's the one I marked but needed to be sure.
Can someone explain to me how you got those values from the (x,y) values ???? help? I want to learn how to do it for myself and my teacher didn't include this part in the lesson. :(((
If we call the vertices A,B,C and the opposite sides a,b,c then we have
(A): a=3, b=4, c=5
(B): a=6, b=√85 stop right there
(C): a=13, b=10√2 stop right there
(D): a=8, b=10, c=6
we have a winner!
If anyone has any tips please let me know
Well, I have to say, triangles have three sides, and the perimeter is just the sum of all the side lengths. So let's add up the lengths of the sides for each option:
A.) (-1,3),(2,-1),(-1,-1)
Side 1: distance between (-1,3) and (2,-1) = sqrt((2 - (-1))^2 + (-1 - 3)^2) = sqrt(9 + 16) = sqrt(25) = 5
Side 2: distance between (2,-1) and (-1,-1) = sqrt((-1 - 2)^2 + (-1 - (-1))^2) = sqrt(9 + 0) = sqrt(9) = 3
Side 3: distance between (-1,-1) and (-1,3) = sqrt((-1 - (-1))^2 + (3 - (-1))^2) = sqrt(0 + 16) = sqrt(16) = 4
Total perimeter = 5 + 3 + 4 = 12, which is not 24.
B.) (6,0),(6,7),(0,7)
Side 1: distance between (6,0) and (6,7) = sqrt((6 - 6)^2 + (7 - 0)^2) = sqrt(0 + 49) = sqrt(49) = 7
Side 2: distance between (6,7) and (0,7) = sqrt((0 - 6)^2 + (7 - 7)^2) = sqrt(36 + 0) = sqrt(36) = 6
Side 3: distance between (0,7) and (6,0) = sqrt((6 - 0)^2 + (0 - 7)^2) = sqrt(36 + 49) = sqrt(85)
Total perimeter = 7 + 6 + sqrt(85)
Hmm... I guess I got a bit carried away with the calculations. But hey, at least we know that adding 7 and 6 will not give us a perimeter of 24. So let's move on!
C.) (-1,-1),(-6,-13),(-9,-9)
Side 1: distance between (-1,-1) and (-6,-13) = sqrt((-6 - (-1))^2 + (-13 - (-1))^2) = sqrt(25 + 144) = sqrt(169) = 13
Side 2: distance between (-6,-13) and (-9,-9) = sqrt((-9 - (-6))^2 + (-9 - (-13))^2) = sqrt(9 + 16) = sqrt(25) = 5
Side 3: distance between (-9,-9) and (-1,-1) = sqrt((-1 - (-9))^2 + (-1 - (-9))^2) = sqrt(64 + 64) = sqrt(128) = 8√2
Total perimeter = 13 + 5 + 8√2
Well, I'm afraid 13 + 5 + 8√2 is not equal to 24. So maybe this option is also not the one.
D.) (-3,-4),(3,-4),(3,4)
Side 1: distance between (-3,-4) and (3,-4) = sqrt((3 - (-3))^2 + (-4 - (-4))^2) = sqrt(36 + 0) = sqrt(36) = 6
Side 2: distance between (3,-4) and (3,4) = sqrt((3 - 3)^2 + (4 - (-4))^2) = sqrt(0 + 64) = sqrt(64) = 8
Side 3: distance between (3,4) and (-3,-4) = sqrt((-3 - 3)^2 + (-4 - 4)^2) = sqrt((-6)^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10
Total perimeter = 6 + 8 + 10 = 24
Bingo! The lengths of the sides for option D) add up to 24 units, which means it could be the vertices of the triangle with a perimeter of exactly 24 units.
So, the answer is D) (-3,-4),(3,-4),(3,4). Tada!
To determine which of the given sets of points could be the vertices of a triangle with a perimeter of 24 units, we need to calculate the distances between the three pairs of points in each set. If the sum of these distances is equal to 24, then the set of points is a valid solution.
To calculate the distance between two points, we can use the distance formula, which is derived from the Pythagorean theorem:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distances for each set of points:
A.) (-1,3), (2,-1), (-1,-1)
- Distance between (-1,3) and (2,-1):
d1 = sqrt((2 - (-1))^2 + (-1 - 3)^2) = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5
- Distance between (-1,3) and (-1,-1):
d2 = sqrt((-1 - (-1))^2 + (-1 - 3)^2) = sqrt(0^2 + (-4)^2) = sqrt(16) = 4
- Distance between (2,-1) and (-1,-1):
d3 = sqrt((-1 - 2)^2 + (-1 - (-1))^2) = sqrt((-3)^2 + 0^2) = sqrt(9) = 3
Total perimeter = d1 + d2 + d3 = 5 + 4 + 3 = 12 units
Since the total perimeter is not equal to 24 units for set A, it is not a valid solution.
Now, let's do the same calculation for the other sets:
B.) (6,0), (6,7), (0,7)
- Distance between (6,0) and (6,7):
d1 = sqrt((6 - 6)^2 + (7 - 0)^2) = sqrt(0^2 + 7^2) = sqrt(49) = 7
- Distance between (6,0) and (0,7):
d2 = sqrt((0 - 6)^2 + (7 - 0)^2) = sqrt((-6)^2 + 7^2) = sqrt(36 + 49) = sqrt(85)
- Distance between (6,7) and (0,7):
d3 = sqrt((0 - 6)^2 + (7 - 7)^2) = sqrt((-6)^2 + 0^2) = sqrt(36) = 6
Total perimeter = d1 + d2 + d3 = 7 + sqrt(85) + 6
Since we do not have exact values for d1 and d2, we cannot determine if the total perimeter equals 24. Therefore, set B cannot be confirmed as a valid solution.
C.) (-1,-1), (-6,-13), (-9,-9)
- Calculate distances between points in set C:
d1 = sqrt((-6 - (-1))^2 + (-13 - (-1))^2) = sqrt((-5)^2 + (-12)^2) = sqrt(25 + 144) = sqrt(169) = 13
d2 = sqrt((-9 - (-6))^2 + (-9 - (-13))^2) = sqrt((-3)^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5
d3 = sqrt((-9 - (-1))^2 + (-9 - (-1))^2) = sqrt((-8)^2 + (-8)^2) = sqrt(64 + 64) = sqrt(128) = 8√2
Total perimeter = d1 + d2 + d3 = 13 + 5 + 8√2
Since we do not have an exact value for d3, we cannot determine if the total perimeter equals 24. Therefore, set C cannot be confirmed as a valid solution.
D.) (-3,-4), (3,-4), (3,4)
- Distance between (-3,-4) and (3,-4):
d1 = sqrt((3 - (-3))^2 + (-4 - (-4))^2) = sqrt((6)^2 + (0)^2) = sqrt(36) = 6
- Distance between (-3,-4) and (3,4):
d2 = sqrt((3 - (-3))^2 + (4 - (-4))^2) = sqrt((6)^2 + (8)^2) = sqrt(36 + 64) = sqrt(100) = 10
- Distance between (3,-4) and (3,4):
d3 = sqrt((3 - 3)^2 + (4 - (-4))^2) = sqrt((0)^2 + (8)^2) = sqrt(64) = 8
Total perimeter = d1 + d2 + d3 = 6 + 10 + 8 = 24 units
Since the total perimeter is equal to 24 units for set D, it can be confirmed as a valid solution.
Based on the calculations, the only set of points that could be the vertices of a triangle with a perimeter of exactly 24 units is option D: (-3,-4), (3,-4), (3,4).