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Standard Level

3.02 Distance formula

Worksheet
Distance between two points
1

Find the length of each vertical interval on the number planes:

a

A \left(2, 5\right) and B \left(2, 8\right)

1
2
3
4
x
1
2
3
4
5
6
7
8
9
y
b

A \left(-7, -1\right) and B \left(-7, -6\right)

-9
-8
-7
-6
-5
-4
-3
-2
-1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
y
c

A \left(4, - 5 \right) and B \left(4, 5\right)

-4
-3
-2
-1
1
2
3
4
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
d

A \left( - 2 , 2\right) and B \left( - 2 , - 3 \right)

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
2

Find the length of each horizontal interval on the number planes:

a

A \left(3, 2\right) and B \left(9, 2\right)

1
2
3
4
5
6
7
8
9
x
1
2
3
4
y
b

A \left( - 4 , 5\right) and B \left( - 7 , 5\right)

-9
-8
-7
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
7
8
9
y
c

A \left( - 5 , - 4 \right) and B \left(9, - 4 \right)

-6
-4
-2
2
4
6
8
10
x
-5
-4
-3
-2
-1
y
d

A \left( 3 , - 1 \right) and B \left(-5, - 1 \right)

-6
-5
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
y
3

For each of the following right-angled triangles, find the length of side AC. Round your answer to two decimal places if necessary.

a
1
2
3
4
5
6
7
x
1
2
3
4
5
6
7
y
b
1
2
3
4
5
x
1
2
3
4
5
y
4

For each of the following right-angled triangles:

i

Find the length of interval PQ.

ii

Find the length of interval QR.

iii

Find the length of PR, rounding to two decimal places if necessary.

a

P \left( - 6 , 5\right), Q \left( - 6 , 2\right) and R \left( - 2 , 2\right)

-7
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
y
b

P \left( - 1 , 9\right), Q \left( - 1 , 6\right) and R \left( - 5 , 6\right)

-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
7
8
9
y
c

P \left( - 4 , 6\right), Q \left( - 4 , 2\right) and R \left(3, 2\right)

-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
y
d

P \left( - 3 , - 6 \right), Q \left( - 3 , - 1 \right) and R \left(1, - 1 \right)

-4
-3
-2
-1
1
2
x
-6
-5
-4
-3
-2
-1
y
5

For each of the following right-angled triangles:

i

Find the length of interval AB.

ii

Find the length of interval BC.

iii

Find the length of AC denoted by c, rounding to two decimal places if necessary.

a

A \left( - 2 , 4\right), B \left( - 2 , - 1 \right) and C \left( - 14 , - 1 \right)

-12
-10
-8
-6
-4
-2
x
-2
-1
1
2
3
4
y
b

A \left( - 2 , 7\right), B \left( - 2 , - 4 \right) and C \left(5, - 4 \right)

-3
-2
-1
1
2
3
4
5
6
x
-4
-2
2
4
6
y
6

Find the length of AB shown on the graph. Round your answer to two decimal places.

-6
-5
-4
-3
-2
-1
1
2
3
4
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
Distance formula
7

Find the distance of the given point P from the origin.

a

P \left(12, 16\right)

2
4
6
8
10
x
2
4
6
8
10
12
14
16
y
b

P \left( - 12 , 16\right)

-12
-10
-8
-6
-4
-2
x
2
4
6
8
10
12
14
16
y
c

P \left(7, 11\right)

1
2
3
4
5
6
7
8
x
2
4
6
8
10
y
d

P \left( - 5 , - 4 \right)

-6
-5
-4
-3
-2
-1
x
-5
-4
-3
-2
-1
y
8

For each of the following graphs, find the length of the interval AC. Round your answer to two decimal places if necessary:

a
1
2
3
4
x
1
2
3
4
5
6
y
b
-5
-4
-3
-2
-1
x
-5
-4
-3
-2
-1
y
c
-4
-3
-2
-1
1
2
3
4
x
1
2
3
4
5
y
d
-2
-1
1
2
3
4
x
1
2
3
4
5
6
7
8
9
y
e
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
-3
-2
-1
1
2
3
4
5
y
f
-2
2
4
6
8
10
x
-7
-6
-5
-4
-3
-2
-1
1
2
3
y
9

Find the distance between Point A and Point B. Write your answer in surd form if necessary.

a

A \left(1, 4\right) and B \left(7, 12\right)

b

A \left(4, 2\right) and B \left( - 8 , - 7 \right)

c

A \left( - 1 , 9\right) and B \left( - 4 , 1\right)

d

A \left(-1, - \dfrac{3}{5} \right) and B \left(4, \dfrac{12}{5}\right)

10

Consider the points M \left( - 9 , - 1 \right) and N \left(1, 5\right).

a

Find the exact distance from M to the origin.

b

Find the exact distance from N to the origin.

c

Which point is closer to the origin?

11

Given P \left(4, 3\right), M \left( - 3 , - 4 \right) and N \left( - 7 , 1\right).

a

Find the distance from P to M to two decimal places.

b

Find the distance from P to N to two decimal places.

c

Which point is further from P?

12

Consider the Points A \left(12, 3\right) and B \left(14, 0\right).

a

Find the length of AB to two decimal places.

b

If M is the midpoint of AB, find the length of AM to two decimal places.

13

AM is a vertical interval 3 units long. If A is the point \left( - 2 , 6\right), find two possible coordinates of M.

14

Consider the points A \left( - 2 , - 6 \right), B \left(4, - 2 \right) and C \left(1, - 4 \right). Find:

a

The exact distance AC.

b

The exact distance BC.

c

What do you notice about the distance between the points?

15

K is the midpoint of A \left(3, 1\right) and C \left(15, - 7 \right). Find the distance from A to K correct to one decimal place.

Geometric applications
16

A triangle is formed by three points:A \left(4, - 3 \right), B\left(1, 0\right) and C \left(7, 0\right). Find the following:

a

The distance BC.

b

The exact distance AB.

c

The exact distance AC.

d

Is this triangle equilateral, isosceles or scalene?

17

ABCD is a rhombus whose vertices are A \left(1, 2\right), B \left(3, 10\right), C \left(11, 12\right) and D \left(9, 4\right). Find:

a

The exact length of diagonals:

i

AC

ii

BD

b

The exact area of the rhombus.

18

A circle with centre at point C \left(3, 4\right) has point A \left( - 12 , 12\right) lying on its circumference. Find:

a

The radius of the circle.

b

The exact circumference of the circle.

c

The distance between point \left(20, 4\right) and the centre.

d

Does the circle also pass through the point \left(20, 4\right)? Explain your answer.

19

A triangle has vertices at A \left(1, - 1 \right), B \left( - 3 , - 4 \right) and C \left(5, - 4 \right).

a

Find the length of the following sides:

i

AB

ii

AC

iii

BC

b

Is this triangle equilateral, isosceles or scalene?

20

Consider the triangle whose vertices are A \left(9, - 2 \right), B \left(2, - 8 \right) and C \left(3, 5\right).

a

How can we show that the triangle has a 90 \degree and two 45 \degree angles?

b

Find the length of the following sides in exact form:

i

AB

ii

AC

iii

BC

c

Does the triangle have a 90 \degree and two 45 \degree angles?

21

\triangle PQR has vertices P \left(2, - 6 \right), \, Q \left( - 9 , - 17 \right) and R \left( - 5 , 1\right).

a

Find the length of the following sides in exact form:

i

PQ

ii

QR

iii

PR

b

Is the triangle right-angled?

22

The points M \left( - 3 , - 5 \right) and R \left(4, - 12 \right) are the end points of an interval. N \left( - 1 , - 7 \right) is a point on this interval.

a

Find the exact distance between the following points:

i

M and N

ii

N and R

b

In what ratio does N divide MR?

23

Find the perimeter of the parallelogram whose vertices are A \left(3, 2\right), B \left( - 2 , 7\right), C \left(2, 9\right) and D \left(7, 4\right). Round your answer correct to one decimal place.

24

The isosceles \triangle PQR is shown on the number plane:

a

Find the area of the triangle.

b

Find the exact length of PR.

c

Find the exact value of d, the perpendicular distance from Q to the side PR.

1
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11
x
1
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9
10
11
y
25

Consider the points P \left( 10 x, - 6 x\right) and Q \left( 4 x, 2 x\right), where x > 0.

a

Find the distance between P and Q in terms of x.

b

Find the coordinates of the midpoint of the segment PQ in terms of x.

26

P \left(- 4, - 4 \right), Q \left(5, - 7 \right), R \left(9 , - 11\right) and S \left(0, - 8\right) are the vertices of a quadrilateral.

a

Find the exact length of side PQ.

b

Find the exact length of side QR.

c

Find the exact length of side RS.

d

Find the exact length of side SP.

e

Find the exact length of the diagonal PR.

f

Find the exact length of the diagonal QS.

g

Classify the quadrilateral.

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