has vertices at A(3,4), B(−1,1), and C(−2,5).
Triangle A B C in the plane with the coordinates as described in the text.
What is the approximate perimeter of △ABC?
15 units
68 units
14.2 units
8.2 units
i thinks it is either 8.2 or 14.2
A(3,4), B(−1,1), and C(−2,5)
A to B
in x = -1 - 3 = -4
in y = 1 - 4 = -3
distance AB = sqrt (3^2+4^2 = 5
B to C
-2 + 1 = -1
5 - 1 = 4
sqrt (17) = 4.12
C to A
-5
1
sqrt 26
5.10
so 5 + 4.12 + 5.10 = 14.22
To find the perimeter of a triangle, we need to calculate the distance between each pair of vertices and then add them up.
First, let's calculate the distance between points A and B:
Distance AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-1 - 3)^2 + (1 - 4)^2)
= sqrt((-4)^2 + (-3)^2)
= sqrt(16 + 9)
= sqrt(25)
= 5
Next, let's calculate the distance between points B and C:
Distance BC = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-2 - (-1))^2 + (5 - 1)^2)
= sqrt((-1)^2 + 4^2)
= sqrt(1 + 16)
= sqrt(17)
Finally, let's calculate the distance between points C and A:
Distance CA = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((3 - (-2))^2 + (4 - 5)^2)
= sqrt((5)^2 + (-1)^2)
= sqrt(25 + 1)
= sqrt(26)
Now, let's calculate the perimeter:
Perimeter = AB + BC + CA
= 5 + sqrt(17) + sqrt(26)
≈ 8.2 units
Therefore, the approximate perimeter of triangle ABC is 8.2 units.
To find the perimeter of the triangle △ABC, you need to calculate the sum of the lengths of its three sides.
Let's calculate the lengths of the sides using the distance formula:
Distance between A and B:
dAB = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-1 - 3)^2 + (1 - 4)^2)
= √((-4)^2 + (-3)^2)
= √(16 + 9)
= √25
= 5
Distance between B and C:
dBC = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-2 - (-1))^2 + (5 - 1)^2)
= √((-1)^2 + 4^2)
= √(1 + 16)
= √17
Distance between C and A:
dCA = √((x2 - x1)^2 + (y2 - y1)^2)
= √((3 - (-2))^2 + (4 - 5)^2)
= √((3 + 2)^2 + (-1)^2)
= √(25 + 1)
= √26
Now, sum up the lengths of all three sides:
Perimeter of △ABC = dAB + dBC + dCA
= 5 + √17 + √26
Approximately, the perimeter is 5 + (√17) + (√26).
Calculating this, you will find that the approximate answer is 14.2 units. Therefore, the correct answer is 14.2 units.