Plot the points A(2,2), B(4,6),C(10,-2)Notice that these points are vertices of a right triangle (the angle is 90 degrees).
Find the area of the triangle ABC:
A(2, 2), B(4, 6), C(10, -2).
(AB)^2 = (4 -2)^2 + (6 -2 )^2,
(AB)^2 = = 4 + 16 = 20.
AB=sqrt20 = sqrt(4*5) = 2*sqrt5 = 4.47.
(BC)^2 = (10 -4)^2 + (-2 -6)^2,
(BC)^2 = 36 + 64 = 100,
BC = sqrt100 = 10.
(AC)^2 = (10 -2)^2 + (-2 -2)^2,
(AC)^2 = 64 + 16 = 80,
AC=sqrt80 = sqrt(16*5) = 4*sqrt5=8.94.
A = AC * AB / 2,
A=4sqrt5 * 2sqrt5/2 = 8 * 5 / 2=20.
To find the area of triangle ABC, we can use the formula for calculating the area of a triangle. The formula is:
Area = (1/2) * base * height
In this case, we can consider either AB or BC as the base of the triangle. Let's consider AB as the base.
To calculate the height of the triangle, we will need to find the length of the perpendicular line drawn from point C to line AB.
To calculate the length of the perpendicular line, we can use the formula for the distance between a point and a line. The formula is:
Distance = |Ax + By + C| / sqrt(A^2 + B^2)
Where (A, B) are the coordinates of a point on the line, and (x, y) are the coordinates of the point to which we want to find the distance.
In this case, let's take point A(2, 2) and point B(4, 6) as two points on line AB. And let's take the coordinates of point C(10, -2) as (x, y).
Now let's calculate the distance:
Distance = |(4-2)(10) + (6-2)(-2) + (2-4)(-2)| / sqrt((4-2)^2 + (6-2)^2)
= |2(10) + 4(-2) - 2(-2)| / sqrt(2^2 + 4^2)
= |20 - 8 + 4| / sqrt(4 + 16)
= |16| / sqrt(20)
= 16 / sqrt(20)
= 16 / (2 * sqrt(5))
= 8 / sqrt(5)
= (8 / sqrt(5)) * (sqrt(5) / sqrt(5))
= (8 * sqrt(5)) / 5
So, the height of the triangle is (8 * sqrt(5)) / 5.
Now we can plug the values into the area formula:
Area = (1/2) * base * height
= (1/2) * AB * ((8 * sqrt(5)) / 5)
= (1/2) * sqrt((4-2)^2 + (6-2)^2) * ((8 * sqrt(5)) / 5)
= (1/2) * sqrt(2^2 + 4^2) * ((8 * sqrt(5)) / 5)
= (1/2) * sqrt(4 + 16) * ((8 * sqrt(5)) / 5)
= (1/2) * sqrt(20) * ((8 * sqrt(5)) / 5)
= (1/2) * (2 * sqrt(5)) * ((8 * sqrt(5)) / 5)
= sqrt(5) * ((8 * sqrt(5)) / 5)
= (8 * sqrt(5) * sqrt(5)) / 5
= (8 * 5) / 5
= 8
Therefore, the area of triangle ABC is 8.