Let A (2, 3); B (-2, -1) and C (5, -2) be vertices of triangle ABC. If the line connected by points B and C is the base of these triangle and the line bisected through point A is the altitude then what will be the area of the triangle?
To find the area of a triangle when you have the coordinates of its vertices, you can use the formula for the area of a triangle given the coordinates of its vertices:
Area = 1/2 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Now, let's calculate the area of triangle ABC using the given coordinates.
First, we need to find the coordinates of the mid-point of the base BC, which are the coordinates of the point where the altitude from A intersects BC.
The mid-point of BC = ( (x₂ + x₃)/2 , (y₂ + y₃)/2 )
Given:
B(-2, -1) and C(5, -2)
Mid-point of BC = ( (-2 + 5)/2 , (-1 + (-2))/2 )
= (3/2 , -3/2)
Now, let's calculate the area of triangle ABC using the formula:
Area = 1/2 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Given:
A(2, 3), B(-2, -1), C(5, -2)
Area = 1/2 * |2(-1 - (-2)) + (-2)(-2 - 3) + 5(3 - (-1))|
= 1/2 * |2(1) + (-2)(-5) + 5(4)|
= 1/2 * (2 + 10 + 20)
= 1/2 * 32
= 16
Therefore, the area of triangle ABC is 16 square units.