Coordinate Geometry Proofs — Practice
Put a shape on axes and every claim about it becomes a calculation you can check.
What these questions cover
These questions use 15 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Bisect — Cut exactly in half.
- Work backwards — Start from the result and undo each step.
- Slope — How much y changes for each 1 that x increases.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Distance formula — d = √(x2 - x1)2 + (y2 - y1)2.
- Midpoint — The point halfway between two others: average the x values and average the y values.
- Parallel — Same slope, different intercept.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
- Simplest radical form — No perfect square factors left under the root.
- Segment — Two endpoints and everything between them. Written AB.
- Prove — The statement you must arrive at. Always the last line.
- Parallelogram — A quadrilateral with both pairs of opposite sides parallel.
- Diagonal — A segment joining two non-adjacent vertices.
- Adjacent — The side next to that angle which is not the hypotenuse.
Problem 1
Find the slope of the segment from (1,2) to (5,4).
Worked answer
1/2
4-25-1 = 24 = 12
Problem 2
Find the length of the segment from (0,0) to (6,8).
Worked answer
10
√36+64 = √100
= 10
Problem 3
Find the midpoint of (2,4) and (8,10).
Worked answer
(5, 7)
(2+82, 4+102) = (5,7)
Problem 4
A line has slope 34. What slope is perpendicular to it?
Worked answer
-4/3
-43
Problem 5
You want to prove two sides are congruent. Which formula do you use?
Worked answer
distance
Congruent means equal length.
Use the distance formula.
Problem 6
You want to prove two sides are parallel. Which formula?
Worked answer
slope
Equal slopes mean parallel.
Problem 7
You want to prove the diagonals bisect each other. Which formula?
Worked answer
midpoint
If both diagonals have the same midpoint, they bisect each other.
Problem 8
Are the segments (1,2)–(5,4) and (3,6)–(7,8) parallel? Answer yes or no.
Worked answer
yes
Both have slope 24 = 12.
Equal slopes, so parallel.
Problem 9
Are the segments (0,0)–(2,4) and (0,0)–(4,-2) perpendicular? Answer yes or no.
Worked answer
yes
Slopes 2 and -12
2 × -12 = -1, so perpendicular.
Problem 10
A quadrilateral has both pairs of opposite sides with equal slopes. What is it?
Worked answer
parallelogram
Both pairs of opposite sides are parallel.
That is a parallelogram.
Problem 11
Find the exact length from (-3,4) to (2,-1) in simplest radical form.
Worked answer
5sqrt(2)
√25+25 = √50
= 5√2
Problem 12
Two sides meet where one is vertical and one is horizontal. Are they perpendicular?
- Yes, though the slope product rule does not apply
- No
- Only if the lengths match
- Cannot be determined
Worked answer
Yes, though the slope product rule does not apply
They meet at a right angle.
But undefined times 0 is not -1, so use the picture instead.
Problem 13
A(1,2), B(5,4), C(7,8), D(3,6). Is ABCD a rectangle? Answer yes or no.
Worked answer
no
Slope AB = 12, slope BC = 2
12 × 2 = 1, not -1
Not a right angle, so not a rectangle.
Problem 14
A(0,0), B(4,0), C(4,3), D(0,3). Find the length of a diagonal.
Worked answer
5
(0,0) to (4,3): √16+9 = 5
Problem 15
Prove the diagonals of the rectangle (0,0), (a,0), (a,b), (0,b) are congruent. Give the common length.
Worked answer
sqrt(a^2 + b^2)
(0,0) to (a,b): √a2+b2
(a,0) to (0,b): √a2+b2
Equal for every a and b.
Problem 16
The midpoint of AB is (4,5) and A is (1,2). Find B.
Worked answer
(7, 8)
1+x2 = 4, so x = 7
2+y2 = 5, so y = 8