Distance on the Coordinate Plane — Drills
Distance on the Coordinate Plane has 7 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 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.
Distance
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the distance between (0, 5) and (12, 14).
Worked answer
15
Run: 12. Rise: 9
122 + 92 = 225
√225 = 15
Problem 2
Find the distance between (-2, -1) and (18, 14).
Worked answer
25
Run: 20. Rise: 15
202 + 152 = 625
√625 = 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
Exact
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the exact distance from (0, 0) to (4, 5), in simplest radical form.
Worked answer
sqrt(41)
42 + 52 = 41
√41 has no square factor, so it is already simplest.
Problem 2
Find the exact distance from (0, 0) to (6, 4), in simplest radical form.
Worked answer
2sqrt(13)
62 + 42 = 52
√52 = √4 · 13 = 2√13
Perimeter
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A right triangle has vertices (0, 0), (5, 0) and (5, 12). Find its perimeter.
Worked answer
30
Legs: 5 and 12
Hypotenuse: √25 + 144 = 13
5 + 12 + 13 = 30
Problem 2
A right triangle has vertices (0, 0), (9, 0) and (9, 40). Find its perimeter.
Worked answer
90
Legs: 9 and 40
Hypotenuse: √81 + 1600 = 41
9 + 40 + 41 = 90
Comparing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Which point is closer to the origin, (7, 5) or (3, 8)?
- Cannot be determined
- (3, 8)
- (7, 5)
- They are the same distance
Worked answer
(3, 8)
(7,5): 49 + 25 = 74
(3,8): 9 + 64 = 73
73 < 74
Problem 2
Which point is closer to the origin, (1, 4) or (7, 7)?
- (1, 4)
- They are the same distance
- (7, 7)
- Cannot be determined
Worked answer
(1, 4)
(1,4): 1 + 16 = 17
(7,7): 49 + 49 = 98
17 < 98
Finding the missing value
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
The distance from (6, -1) to (6, k) is 4, with k > -1. Find k.
Worked answer
3
|k - (-1)| = 4
Since k > -1, k = -1 + 4
k = 3
Problem 2
The distance from (-7, -3) to (-7, k) is 9, with k > -3. Find k.
Worked answer
6
|k - (-3)| = 9
Since k > -3, k = -3 + 9
k = 6
Shape
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A square has vertices (0,0), (6,0), (6,6), (0,6). Find the length of a diagonal in simplest radical form.
Worked answer
6sqrt(2)
From (0,0) to (6,6)
36 + 36 = 72
√72 = 6√2
Problem 2
A square has vertices (0,0), (8,0), (8,8), (0,8). Find the length of a diagonal in simplest radical form.
Worked answer
8sqrt(2)
From (0,0) to (8,8)
64 + 64 = 128
√128 = 8√2