Coordinate Geometry Proofs — Drills
Coordinate Geometry Proofs has 9 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Slope
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the slope of the segment from (0, 3) to (7, -5).
Worked answer
-8/7
Rise = -8, run = 7
m = -87
Problem 2
Find the slope of the segment from (3, 0) to (6, 7).
Worked answer
7/3
Rise = 7, run = 3
m = 73
Distance
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the length of the segment from (0, 5) to (9, 17).
Worked answer
15
Legs 9 and 12
√81 + 144 = 15
Problem 2
Find the length of the segment from (-2, -1) to (13, 19).
Worked answer
25
Legs 15 and 20
√225 + 400 = 25
Midpoint
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the midpoint of (0, 1) and (14, -1).
Worked answer
(7, 0)
x: 0 + 142 = 7
y: 1 + (-1)2 = 0
Problem 2
Find the midpoint of (5, -2) and (13, -10).
Worked answer
(9, -6)
x: 5 + 132 = 9
y: -2 + (-10)2 = -6
Perpendicular lines
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A line has slope -87. What slope is perpendicular to it?
Worked answer
7/8
Reciprocal: -78
Negated: 78
Check: the product is -1 ✓
Problem 2
A line has slope 13. What slope is perpendicular to it?
Worked answer
-3
Reciprocal: 3
Negated: -3
Check: the product is -1 ✓
Tool
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
You want to prove the shape is a rhombus. Which formula do you use?
Worked answer
distance
"the shape is a rhombus" is a claim about length.
So use the distance formula.
Problem 2
You want to prove two sides are congruent. Which formula do you use?
Worked answer
distance
"two sides are congruent" is a claim about length.
So use the distance formula.
Parallel lines
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Are the segments (5, -1)–(7, 5) and (0, 0)–(2, 6) parallel? Answer yes or no.
Worked answer
yes
First slope: 3
Second slope: 3
Equal, so parallel.
Problem 2
Are the segments (-1, -3)–(1, 5) and (0, 0)–(2, 8) parallel? Answer yes or no.
Worked answer
yes
First slope: 4
Second slope: 4
Equal, so parallel.
Shape
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A quadrilateral is found to have: opposite sides are parallel and adjacent slopes multiply to -1. Name it.
Worked answer
rectangle
Opposite sides are parallel and adjacent slopes multiply to -1.
That is a rectangle.
Problem 2
A quadrilateral is found to have: exactly one pair of sides has equal slopes. Name it.
Worked answer
trapezoid
Exactly one pair of sides has equal slopes.
That is a trapezoid.
Vertical
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
One side is vertical and the next is horizontal. Are they perpendicular?
- Yes, though the slope product rule does not apply
- Cannot be determined
- No
- Only if the lengths match
Worked answer
Yes, though the slope product rule does not apply
They do meet at a right angle.
But a vertical slope is undefined, so m1 m2 = -1 cannot be used — check for vertical sides first.
Proof
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Using convenient coordinates (0,0), (a,0), (a,a), (0,a), prove the diagonals of any square are congruent. Give the common diagonal length.
Worked answer
sqrt(2)a
Compute both diagonals with the distance formula.
Both come to √(2)a.
The letters were never pinned down, so this covers every such shape.
Problem 2
Using convenient coordinates (0,0), (a,0), (a,b), (0,b), prove the diagonals of any rectangle are congruent. Give the common diagonal length.
Worked answer
sqrt(a^2 + b^2)
Compute both diagonals with the distance formula.
Both come to √(a2 + b2).
The letters were never pinned down, so this covers every such shape.