Math 1 · Unit 4 · Lesson 4 · part 3 of 5
Two proofs, worked
Corners A(−3, 1), B(1, 3), C(3, −1), D(−1, −3), in order around the shape.
Two proofs, worked
Worked: is it a rectangle?
Corners A(−3, 1), B(1, 3), C(3, −1), D(−1, −3), in order around the shape.
Slope of AB: (3 − 1)/(1 − (−3)) = 24 = 12. Slope of BC: (−1 − 3)/(3 − 1) = −4/2 = −2. Slope of CD: (−3 − (−1))/(−1 − 3) = −2/−4 = 12. Slope of DA: (1 − (−3))/(−3 − (−1)) = 4/−2 = −2.
Opposite sides: AB and CD both 12, BC and DA both −2. Parallel. Neighbors: 12 × (−2) = −1 at every corner. Right angles.
ABCD is a rectangle.
Worked: is it a right triangle?
Corners P(−2, −1), Q(4, 2), R(2, 6).
Slope of PQ: (2 − (−1))/(4 − (−2)) = 36 = 12. Slope of QR: (6 − 2)/(2 − 4) = 4/−2 = −2. Slope of RP: (−1 − 6)/(−2 − 2) = −7/−4 = 74.
PQ and QR: 12 × (−2) = −1. Perpendicular, so the corner at Q is a right angle.
PQR is a right triangle, with the right angle at Q. (The third side is neither parallel nor perpendicular to the others, which is fine: one right angle is all a right triangle needs.)