Coordinate Geometry Proofs — Challenge
Put a shape on axes and every claim about it becomes a calculation you can check.
What these problems 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
What is the midpoint of the segment joining (0,0) and (6,8)?
- (3,4)
- (6,8)
- (4,3)
- (1.5,2)
- (12,16)
Worked answer
(3,4)
Average the x values and the y values separately.
0+62 = 3 and 0+82 = 4.
So the midpoint is (3,4).
Problem 2
What is the slope of the segment from (1,1) to (4,7)?
- 1
- 2
- 3
- 6
- 0.5
Worked answer
2
The change in y is 7-1 = 6 and the change in x is 4-1 = 3.
The slope is 63 = 2.
Problem 3
Two segments are parallel. What is true of their slopes?
- They are negative reciprocals
- They are equal
- They are reciprocals
- They are opposite in sign
- They are both zero
Worked answer
They are equal
Slope measures steepness and direction, and parallel segments point the same way.
So their slopes are equal.
Negative reciprocals describe perpendicular segments instead.
Problem 4
Two segments are perpendicular. What do their slopes multiply to give?
- -1
- 0
- 1
- 2
- It is undefined
Worked answer
-1
Perpendicular slopes are negative reciprocals of one another, such as 2 and -12.
Multiplying a number by its negative reciprocal gives -1: 2·(-12) = -1.
The one exception is a vertical and a horizontal segment, where one slope is undefined.
Problem 5
A quadrilateral has both pairs of opposite sides parallel. What is it?
- A trapezium
- A kite
- A parallelogram
- A rhombus
- A square
Worked answer
A parallelogram
Both pairs of opposite sides parallel is the definition of a parallelogram.
Using coordinates, this is shown by checking that opposite sides have equal slopes.
A rhombus or a square would need the extra condition of equal side lengths, which is not given.
Problem 6
What is the length of the segment from (0,0) to (5,12)?
- 17
- 13
- 60
- 169
- 7
Worked answer
13
Use the distance formula, which is Pythagoras applied to the coordinate gaps.
√52+122 = √25+144 = √169 = 13.
Adding the coordinates gives 17 and forgetting the root gives 169; both are among the options.
Problem 7
To prove a triangle isosceles using coordinates, what should be shown?
- Two sides have equal length
- All slopes are equal
- The area is zero
- The midpoints coincide
- All angles are 60 degrees
Worked answer
Two sides have equal length
Isosceles means two sides of equal length.
With coordinates, lengths come from the distance formula, so compute all three side lengths and show two agree.
Equal slopes would mean the points are in a straight line, which gives no triangle at all.
Problem 8
A triangle has vertices (0,0), (4,0) and (0,3). What kind of triangle is it?
- Equilateral
- Obtuse
- Right-angled
- Isosceles
- Degenerate
Worked answer
Right-angled
Two of the sides run along the axes: one from (0,0) to (4,0) is horizontal, and one from (0,0) to (0,3) is vertical.
A horizontal and a vertical segment meet at a right angle, so the angle at the origin is 90 degrees.
The sides are 3, 4 and 5, so the triangle is right-angled and not isosceles.