Cogito
Geometry · Chapter 7 · Lesson 3
Proving Things with Coordinates
Turning a geometric claim into arithmetic.
11 problems · about 23 minutes · G-GPE.B.4, G-GPE.B.5
What this lesson teaches
The student proves a property of a figure using distance, midpoint, and slope.
- Placing a figure on axes turns geometry into arithmetic.
- Distance settles lengths, slope settles direction, midpoint settles bisection.
- Using letters rather than numbers is what makes it a proof.
Warm Up
Straightforward practice. Get the method working first.
4 problemsFrom (2, 1) to (6, 4). Distance?
Answer 5
Why 5.
From (1, 2) to (4, 6). Distance?
Answer 5
Why 3 and 4.
Midpoint of (2, 4) and (8, 10). What is its y?
Answer 7
Why Average 4 and 10.
Slopes 3 and 3. Parallel? 1 for yes, 0 for no.
Answer 1
Why Equal slopes.
Build It Up
The same ideas with more to keep track of.
3 problemsSlopes 2 and −0.5. Perpendicular? 1 for yes, 0 for no.
Answer 1
Why Product is −1.
Match each claim to the tool that settles it.
Answer These sides are equal → Distance formula; These sides are parallel → Slope; This point bisects the segment → Midpoint formula
Why Only one of the three is about direction.
From (0, 0) to (5, 12). Distance?
Answer 13
Why 25 + 144.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsDoes using letters instead of numbers make a proof general? 1 for yes, 0 for no.
Answer 1
Why It covers every case.
Midpoint of (−4, 0) and (4, 0). What is its x?
Answer 0
Why They balance.
The Distance: From (0, 0) to (3, 4). Distance?
Answer 5
Why 5.
The Midpoint: Midpoint of (0, 0) and (8, 4). What is its x?
Answer 4
Why 4.