Find the missing side lenghts, perimeter and area of triangle ABC on the coordinates plane. All answers should be in simplest radical form, if necessary.

Question

.A (-1,4). B (-3,2) C (0,2)
Side length AB is?
Side length BC is?
Side length AC is?
Perimeter of triangle ABC is?
Area of triangle ABC is?

Answer 1

To find the lengths of the sides AB, BC, and AC, we can use the distance formula:

Distance formula: √((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we can find the lengths of the sides:

AB: √((-3 - (-1))^2 + (2 - 4)^2) = √((-3 + 1)^2 + (2 - 4)^2) = √((2)^2 + (-2)^2) = √(4 + 4) = √8 = 2√2

BC: √((0 - (-3))^2 + (2 - 2)^2) = √((0 + 3)^2 + (0)^2) = √((3)^2 + (0)^2) = √9 = 3

AC: √((0 - (-1))^2 + (2 - 4)^2) = √((0 + 1)^2 + (2 - 4)^2) = √((1)^2 + (-2)^2) = √(1 + 4) = √5

So, the lengths of the sides are:

AB = 2√2
BC = 3
AC = √5

To find the perimeter, we simply add up the lengths of the sides:

Perimeter = AB + BC + AC = 2√2 + 3 + √5

To find the area of the triangle, we can use the formula for the area of a triangle:

Area = 0.5 * base * height

The base of the triangle can be found by finding the horizontal distance between points A and B, which is 2. The height can be found by finding the vertical distance from point B to the line AC, which is 2. So:

Area = 0.5 * base * height = 0.5 * 2 * 2 = 2

Therefore, the answers are:
AB = 2√2
BC = 3
AC = √5
Perimeter of triangle ABC = 2√2 + 3 + √5
Area of triangle ABC = 2.