Question 181783: Find the area of the triangle with the given vertices:
A (-2,-6)
B (-2,-2) ,and
C (8,1)
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
First plot your points. Then notice that you can construct an altitude for this triangle from (8,1) to (-2,1), if you consider the base of the triangle to be the segment AB.
Since your altitude segment is parallel to the x-axis, you can determine the distance between (8,1) and (-2,1) by inspection -- it is simply |8 - (-2)| = 10.
Similarly, you can calculate the length of the base, AB, directly |-6 - (-2)| = 4.
Then just use the area of a triangle formula: 
On the other hand, if you are really into pain, you could calculate the altitude considering AC as the base segment.
First, use the distance formula to calculate the length of the base. Then use the two-point form of the equation of a line and points A and C to determine the equation of the line in which segment AC lies. Find the slope of that line and take the negative reciprocal to determine the slope of a line perpendicular to the base. Then use the point-slope form of the equation of a line to find the equation of the perpendicular to AC through B. Solve the system of equations to find the point of intersection of the altitude to the base. Use the distance formula to calculate the distance from the point of intersection of the altitude and the base to point B giving you the length measure of the altitude. Use the two length measures in the formula for the area of a triangle. If you made no errors, you should come up with 20 square units. I tried it this way first, and half-way in said "Duh!" Believe me, this is a computational horror I wouldn't wish on someone I didn't like.
John
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