Unit 4 Test, Part 2: Analytic Geometry

Question

1.   The coordinates of the vertices of ∆PQR are P(5, -2), Q(1, -1), and R(3, 7). Determine whether ∆PQR is a right triangle. Use slopes to show this and graph this triangle or use www.desmos.com and copy image below.

Answer 1

If you quickly sketch the points, the two sides that appear to be most likely perpendicular to one another are side RQ and QP. So, simply find the slope of each of these line segments:

For RQ, m = (7 - (-1))/(3 - 1) = 8/2 = 4

For QP, m = (-2 +1)/(5 - 1) = -1/4

Notice that 4 and -1/4 are negative reciprocals of one another, which is key to lines being perpendicular to one another: their slopes must be negative reciprocals. Also, we know that perpendicular lines form 90 degree angles at their intersection. Finally, a right triangle has one angle which is 90 degrees, thus the triangle is a right triangle. :-)