Geometry Chapter 4.7 Triangles and Coordinate Proof
A coordinate proof involves placing geometric figures in a coordinate plane. Then you will use:
1) Distance Formula - to show segments are congruent
2) Midpoint formula - to show that the segment is bisected
3) Slope formula -
a) if 2 lines have the same slope, then the lines are parallel
b) if 2 lines have slopes that are negative reciprocals of each other, then the lines are perpendicular
Example: To show that ΔABC is an isosceles right triangle given A(0,0), B(0,6) and C(6,0).
1. slope of AB = (0-6)/(0-0) = undefined
2. slope of BC = (6 - 0)/(0 - 6) = 6/(-6) = -1
3. slope of AC = (0 - 0)/(0 - 6) = 0 / 6 = 0
Therefore AB is perpendicular to AC because horizontal and vertical lines are perpendicular to each other. therefore angle A is a right angle because perpendicular lines form right angles.
4. AB = 6, AC = 6, and BC = 6√2
so we can conclude that AB = AC because they have the same measure.
Therefore since ΔABC has one right angle and 2 congruent sides, it is an isosceles right triangle.
