Coordinate Proof

Proving geometry with algebra.

12 problems · about 22 minutes · G-GPE.B.4

What this lesson teaches

The student proves geometric statements using coordinates, distance, midpoint and slope.

A coordinate proof places a figure on a grid and argues with algebra.
Distance proves lengths, slope proves parallel or perpendicular, midpoint proves bisection.
Keep the placement convenient and the coordinates general.

Straightforward practice. Get the method working first.

5 problems

A rectangle is 8 by 15. What is the diagonal length?
Answer 17
Why 17.
Why use letters rather than specific numbers in a coordinate proof?
Answer One example does not prove a general statement.
Why A proof must cover every case.
To prove two sides equal, which tool? 1 distance, 2 slope, 3 midpoint.
Answer 1
Why Lengths.
To prove a segment is bisected, which tool? 1, 2 or 3?
Answer 3
Why The halfway point.
To prove a right angle, which tool? 1, 2 or 3?
Answer 2
Why Slopes multiplying to −1.

The same ideas with more to keep track of.

3 problems

A rectangle is 5 by 12. Diagonal length?
Answer 13
Why √169.
Midpoint of (0, 0) and (6, 8). x-coordinate?
Answer 3
Why (0 + 6) ÷ 2.
Does proving it for one 3 by 4 rectangle prove it for all? 1 yes, 0 no.
Answer 0
Why A proof must be general.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each claim with the tool that proves it.
Answer Two sides have equal length → Distance formula; Two sides are parallel → Slope; A diagonal is bisected → Midpoint formula
Why Each tool answers a different kind of question.
A rectangle is 9 by 12. Diagonal length?
Answer 15
Why √225.
The Diagonal: A rectangle is 6 by 8. How long is each diagonal?
Answer 10
Why 10.
The Tool: To prove two sides are parallel, which tool? 1 distance, 2 slope, 3 midpoint.
Answer 2
Why Slope.