To the nearest tenth, find the perimeter of ABC with vertices A(-2-2), B(0,5) and C(3,-1).
19.1 units
AB= 7.3
BC=6.8
CA=5
AB=sqrt{(x₂-x₁)²+(y₂-y₁)²} =
=sqrt{(0-(-2))²+(5-(-2)) ²} =
=sqrt(4+49)=7.3
BC = sqrt{(x₃-x₂ )²+(y₃-y₂)²} =
=sqrt{(3-0)²+(-1-5) ²} =
=sqrt(9+36)=6.7
CA = sqrt{(x₁-x₃ )²+(y₁-y₃)²} =
=sqrt{(-2-3)²+(-2+1) ²} =
=sqrt(25+1)=5.1
P=AB+BC+CA = 7.3+6.7+5.1 =14.1
To find the perimeter of triangle ABC, we need to calculate the distances between the three vertices.
First, let's find the distance between points A and B.
Using the distance formula:
d(A, B) = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁, y₁) = (-2, -2) and (x₂, y₂) = (0, 5).
Plugging in the values:
d(A, B) = √[(0 - (-2))² + (5 - (-2))²]
= √[2² + 7²]
= √[4 + 49]
= √53
≈ 7.3 (to the nearest tenth)
Now, let's find the distance between points B and C.
Using the distance formula:
d(B, C) = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁, y₁) = (0, 5) and (x₂, y₂) = (3, -1).
Plugging in the values:
d(B, C) = √[(3 - 0)² + (-1 - 5)²]
= √[3² + (-6)²]
= √[9 + 36]
= √45
≈ 6.7 (to the nearest tenth)
Finally, let's find the distance between points C and A.
Using the distance formula:
d(C, A) = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁, y₁) = (3, -1) and (x₂, y₂) = (-2, -2).
Plugging in the values:
d(C, A) = √[(-2 - 3)² + (-2 - (-1))²]
= √[(-5)² + (-1)²]
= √[25 + 1]
= √26
≈ 5.1 (to the nearest tenth)
Now, we can find the perimeter by adding up the three distances:
Perimeter = d(A, B) + d(B, C) + d(C, A)
≈ 7.3 + 6.7 + 5.1
≈ 19.1 (to the nearest tenth)
Therefore, the perimeter of triangle ABC is approximately 19.1 units (to the nearest tenth).
To find the perimeter of triangle ABC, we need to calculate the distance between each pair of vertices and then add them up.
Let's start by finding the distance between points A and B.
The distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For points A(-2, -2) and B(0, 5), the distance can be calculated as:
Distance_AB = sqrt((0 - (-2))^2 + (5 - (-2))^2)
= sqrt(2^2 + 7^2)
= sqrt(4 + 49)
= sqrt(53)
Now, let's find the distance between points B and C.
Distance_BC = sqrt((3 - 0)^2 + (-1 - 5)^2)
= sqrt(3^2 + (-6)^2)
= sqrt(9 + 36)
= sqrt(45)
Lastly, we need to find the distance between points C and A.
Distance_CA = sqrt((-2 - 3)^2 + (-2 - (-1))^2)
= sqrt((-5)^2 + (-1 + 2)^2)
= sqrt(25 + 1)
= sqrt(26)
Now, let's add up all the distances to find the perimeter of triangle ABC:
Perimeter = Distance_AB + Distance_BC + Distance_CA
= sqrt(53) + sqrt(45) + sqrt(26)
To find the perimeter to the nearest tenth, we can evaluate the above expression using a calculator or use an approximation technique. Let's use a calculator to find the exact value:
Perimeter ≈ 0.4 + 6.7 + 5.1
≈ 12.2
Therefore, to the nearest tenth, the perimeter of triangle ABC is approximately 12.2 units.