Distance on the Coordinate Plane — Practice
Draw the right triangle and the distance formula stops being something to memorise.
What these questions cover
These questions use 11 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Base — The number being multiplied. In 53 the base is 5.
- Work backwards — Start from the result and undo each step.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Leg — One of the two sides forming the right angle.
- Hypotenuse — The side opposite the right angle — always the longest.
- Midpoint — The point halfway between two others: average the x values and average the y values.
- Simplest radical form — No perfect square factors left under the root.
- Segment — Two endpoints and everything between them. Written AB.
- Diagonal — A segment joining two non-adjacent vertices.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
Find the distance between (0, 0) and (3, 4).
Worked answer
5
32 + 42 = 25
√25 = 5
Problem 2
Find the distance between (1, 1) and (7, 9).
Worked answer
10
Run: 6. Rise: 8
36 + 64 = 100
√100 = 10
Problem 3
Find the distance between (2, 5) and (2, 12).
Worked answer
7
It is a vertical segment
12 - 5 = 7
No square root needed
Problem 4
Find the midpoint of (2, 4) and (8, 10).
Worked answer
(5, 7)
x: 2 + 82 = 5
y: 4 + 102 = 7
Problem 5
Find the midpoint of (-3, 6) and (5, -2).
Worked answer
(1, 2)
x: -3 + 52 = 1
y: 6 - 22 = 2
Problem 6
Find the distance between (-2, 3) and (4, 11).
Worked answer
10
Run: 4 - (-2) = 6
Rise: 11 - 3 = 8
√36 + 64 = 10
Problem 7
Find the exact distance between (1, 2) and (4, 6)... then between (0, 0) and (2, 3).
Worked answer
sqrt(13)
22 + 32 = 13
d = √13
Leave it exact
Problem 8
Find the exact distance between (0, 0) and (5, 5), in simplest radical form.
Worked answer
5sqrt(2)
25 + 25 = 50
√50 = √25 · 2
= 5√2
Problem 9
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
Problem 10
A triangle has vertices (0, 0), (6, 0) and (6, 8). Find its perimeter.
Worked answer
24
Base: 6. Height: 8
Hypotenuse: √36 + 64 = 10
6 + 8 + 10 = 24
Problem 11
Which point is closer to the origin, (3, 4) or (1, 5)?
- (3, 4)
- (1, 5)
- They are the same distance
- Cannot be determined
Worked answer
(3, 4)
(3,4): 9 + 16 = 25
(1,5): 1 + 25 = 26
25 < 26, so (3, 4) is closer
Problem 12
Find the distance between (-5, -2) and (-1, 1).
Worked answer
5
Run: -1 - (-5) = 4
Rise: 1 - (-2) = 3
√16 + 9 = 5
Problem 13
The distance from (2, 3) to (2, k) is 9, with k > 3. Find k.
Worked answer
12
|k - 3| = 9
k - 3 = 9 since k > 3
k = 12
Problem 14
A quadrilateral has vertices (0,0), (4,0), (4,4), (0,4). Find the length of a diagonal in simplest radical form.
Worked answer
4sqrt(2)
From (0,0) to (4,4)
16 + 16 = 32
√32 = 4√2
Problem 15
The midpoint of a segment is (0, 0) and one endpoint is (-6, 8). Find the other endpoint.
Worked answer
(6, -8)
-6 + x2 = 0, so x = 6
8 + y2 = 0, so y = -8
Problem 16
Find the perimeter of the triangle with vertices (1, 1), (4, 5) and (1, 5).
Worked answer
12
(1,1) to (1,5): 4
(1,5) to (4,5): 3
(1,1) to (4,5): √9+16 = 5
4 + 3 + 5 = 12