Math 1 · Unit 11 · Lesson 4 · part 3 of 4
The missing corner, and how to write a proof
Rectangle ABCD has A(1, 2), B(6, 0), and C(10, 10).
The missing corner, and how to write a proof
Rectangle ABCD has A(1, 2), B(6, 0), and C(10, 10). Where is D? Use what you know about rectangles: the diagonals of a rectangle (of any parallelogram) share the same midpoint. The diagonals are AC and BD.
- 1Midpoint of AC.(1 + 102, 2 + 102) = (5.5, 6).
- 2That is also the midpoint of BD.So (5.5, 6) is the midpoint of B(6, 0) and the unknown D. This is the missing endpoint question from Lesson 2.
- 3Step way.From B(6, 0) to (5.5, 6) is 0.5 left and 6 up. Once more: (5.5 − 0.5, 6 + 6) = D(5, 12).
- 4Check a right angle.Slope of AB: 0 − 26 − 1 = −25. Slope of BC: 10 − 010 − 6 = 104 = 52. They multiply to −1: angle B is a right angle.
A proof is a few clear sentences. It does not have to be a two-column table. Name the tool and the result, like this: "AB and BC have slopes −25 and 52, which multiply to −1, so angle B is a right angle." Or: "Every side is √13, so the four sides are equal." A reader should be able to check each sentence.
One more, a triangle: A(0, 0), B(6, 0), C(3, 4).