triangle with vertices A(-2,2), B(4,0) and C(-2,-3).
find the length of this altitude
To find the length of the altitude of a triangle, we need to first find the length of the side of the triangle which the altitude is drawn from.
Let's label the vertices of the triangle as follows:
A(-2,2)
B(4,0)
C(-2,-3)
To find the length of side AB, we can use the distance formula:
Distance AB = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of the points A(-2,2) and B(4,0):
Distance AB = √((4 - (-2))^2 + (0 - 2)^2)
= √((6)^2 + (-2)^2)
= √(36 + 4)
= √40
= 2√10
So, the length of side AB is 2√10.
Now, to find the length of the altitude, we'll need to draw the altitude from one vertex of the triangle to the side opposite to it.
Let's draw an altitude from point A to side BC.
The length of the altitude can be found using the formula:
Length of altitude = (2 * Area of the triangle) / Length of the base
In our case, the length of the base is side BC.
To find the area of the triangle, we'll use the formula:
Area of a triangle = 1/2 * base * height
In this case, base = BC and height = length of the altitude.
Let's calculate the area first:
Area = 1/2 * BC * Length of the altitude
We already found the length of side AB, which is the base of the triangle. So, side BC is the side opposite to it.
We can use the distance formula again to find the length of BC:
Distance BC = √((-2 - 4)^2 + (-3 - 0)^2)
= √((-6)^2 + (-3)^2)
= √(36 + 9)
= √45
= 3√5
Now, substituting the values into the area formula:
Area = 1/2 * 3√5 * Length of the altitude
We need to find the length of the altitude, so rearrange the formula:
Length of the altitude = (2 * Area) / (3√5)
Now, let's find the area using the coordinates of the triangle's vertices:
Area = 1/2 * |((x1 * (y2 - y3)) + (x2 * (y3 - y1)) + (x3 * (y1 - y2)))|
= 1/2 * |((-2 * (0 - (-3))) + (4 * (-3 - 2)) + (-2 * (2 - 0)))|
= 1/2 * |((-2 * 3) + (4 * -5) + (-2 * 2))|
= 1/2 * |(-6 - 20 - 4)|
= 1/2 * |-30|
= 15
Now, substitute the area value into the formula for the length of the altitude:
Length of the altitude = (2 * 15) / (3√5)
= 30 / (3√5)
= 10 / √5
= 10√5 / 5
= 2√5
So, the length of the altitude in the given triangle is 2√5 (approximately 4.47 units).