Worksheet: Distance between Two Points on a Coordinate Plane

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In this worksheet, we will practice finding the distance between two points on the coordinate plane using the Pythagorean theorem.

Q1:

Find the distance between the point ( − 2 , 4 ) and the point of origin.

  • A 2 √ 2 length units
  • B √ 6 length units
  • C 3 √ 2 length units
  • D 2 √ 5 length units

Q2:

Find the distance between the points 𝐴 and 𝐵 .

  • A4 length units
  • B √ 1 0 length units
  • C 2 √ 1 3 length units
  • D √ 5 8 length units
  • E16 length units

Q3:

Consider the two points 𝐴 ( 𝑥 , 𝑦 ) and 𝐵 ( 𝑥 , 𝑦 ) .

Using the Pythagorean theorem, find an expression for the length of 𝐴 𝐵 .

  • A √ ( 𝑥 − 𝑥 ) + ( 𝑦 − 𝑦 )
  • B ( 𝑥 + 𝑥 ) − ( 𝑦 + 𝑦 )
  • C ( 𝑥 − 𝑥 ) + ( 𝑦 − 𝑦 )
  • D ( 𝑥 − 𝑥 ) + ( 𝑦 − 𝑦 )
  • E ( 𝑥 + 𝑥 ) − ( 𝑦 + 𝑦 )

Using the distance formula, find the distance between the points ( 3 , 4 ) and ( 5 , 6 ) . Give your answer as a surd in its simplest form.

  • A 2 √ 2
  • B8
  • C2
  • D √ 2
  • E6

Q4:

What is the distance between the point 𝐴 and the origin?

  • A 2 √ 2 length units
  • B √ 6 length units
  • C 3 √ 2 length units
  • D 2 √ 5 length units

Q5:

If the distance between the two points ( 𝑎 , 0 ) and ( − 𝑎 + 1 , 0 ) is 9, find all possible values of 𝑎 .

  • A 𝑎 = 4 or 𝑎 = − 5
  • B 𝑎 = 4 or 𝑎 = 5
  • C 𝑎 = 5 or 𝑎 = − 5
  • D 𝑎 = − 4 or 𝑎 = 5
  • E 𝑎 = − 4 or 𝑎 = 4

Q6:

Which of the following points is at a distance of 5 √ 2 from the origin?

  • A ( 5 , 0 )
  • B ( 0 , 5 )
  • C 5 √ 2 , 5 √ 2
  • D ( 5 , 5 )

Q7:

Given 𝐴 ( 4 , 5 ) , 𝐵 ( 5 , 5 ) , and 𝐶 ( − 4 , − 7 ) , what is the perimeter of △ 𝐴 𝐵 𝐶 ?

  • A18
  • B 1 6 + 2 √ 1 0
  • C 1 6 + 2 √ 5
  • D 1 6 + 4 √ 1 3

Q8:

How far is the butcher from the garage?

Q9:

Find the distance between the points ( 4 , 5 ) and ( 6 , − 2 ) .

  • A √ 1 1 length units
  • B3 length units
  • C √ 5 1 length units
  • D √ 5 3 length units
  • E13 length units

Q10:

The distance between ( 𝑎 , 5 ) and ( 1 , 1 ) is 5. What are the possible values of 𝑎 ?

  • A 𝑎 = 2 or − 4
  • B 𝑎 = 2 or 4
  • C 𝑎 = − 2 or − 4
  • D 𝑎 = − 2 or 4

Q11:

If 𝐴 ( − 7 , 𝑥 ) and 𝐵 ( 9 , 1 4 ) , where 𝐴 𝐵 = 4 √ 1 7 length units, find all the possible values of 𝑥 .

  • A 𝑥 = 1 0 or 𝑥 = − 1 8
  • B 𝑥 = − 1 0 or 𝑥 = − 1 8
  • C 𝑥 = − 1 0 or 𝑥 = 1 8
  • D 𝑥 = 1 0 or 𝑥 = 1 8

Q12:

Two baseball posts were positioned at ( 9 , 3 ) and ( 6 , 9 ) . Find the distance between the posts giving the answer to one decimal place.

Q13:

Nabil is making a map of his local area measured in yards. The coffee shop is at ( − 5 , − 4 ) and the Italian restaurant is at ( 0 , 6 ) . Find the distance between the coffee shop and the Italian restaurant giving the answer to one decimal place.

Q14:

The point 𝑃 is located on the line segment between point 𝐴 ( − 6 , 5 ) and point 𝐷 ( 9 , 8 ) . Given that the distance from 𝐴 to 𝑃 is twice the distance from 𝑃 to 𝐷 , find the coordinates of point 𝑃 .

  • A ( 1 , 4 . 3 3 )
  • B ( 2 , 8 . 6 7 )
  • C ( 3 , 1 3 )
  • D ( 4 , 7 )
  • E ( 1 0 , 1 0 )

Q15:

A small craft in Lake Ontario sends out a distress signal from the coordinates ( 4 9 , 6 4 ) . One rescue boat is at the coordinates ( 6 0 , 8 2 ) , while a Coast Guard boat one is at the coordinates ( 5 8 , 4 7 ) . Assuming both boat travel at the same rate, which one will get to the distressed boat first?

  • AThe coast guard craft
  • BThe rescue boat

Q16:

On a map, coordinates are expressed in miles. The coordinates for San Francisco are ( 5 3 , 1 7 ) and those for Sacramento are ( 1 2 3 , 7 8 ) . Find the distance between the cities to the nearest mile.

Q17:

On a map, each coordinate represents a distance in miles from some fixed point. San Jose’s coordinates are (76, − 1 2 ), while those for San Francisco are (53, 17). Find the distance between San Jose and San Francisco to the nearest mile.

Q18:

Points 𝐴 and 𝐵 have coordinates 𝐴 ( 𝑥 , 𝑦 ) and 𝐵 ( 𝑥 , 𝑦 ) respectively. Point 𝐶 lies on 𝐴 𝐵 such that 𝐴 𝐶 and 𝐵 𝐶 are in the ratio of 𝑘 ∶ 𝑚 .

Write the quotient of the length of 𝐴 𝐶 by the length of 𝐴 𝐵 in terms of 𝑘 and 𝑚 .

  • A 𝑘 + 𝑚 𝑘
  • B 𝑘 𝑚
  • C 𝑚 𝑘
  • D 𝑘 𝑘 + 𝑚
  • E 𝑚 𝑘 + 𝑚

Find an expression for the difference between the 𝑥 coordinates of 𝐴 and 𝐵 .

  • A 𝑥 − 𝑥
  • B 𝑥 𝑥
  • C 𝑥 + 𝑥 2
  • D 𝑥 − 𝑥 2
  • E 𝑥 𝑥

Find an expression for the difference between the 𝑦 coordinates of 𝐴 and 𝐵 .

  • A 𝑦 𝑦
  • B 𝑦 − 𝑦 2
  • C 𝑦 − 𝑦
  • D 𝑦 + 𝑦 2
  • E 𝑦 𝑦

Find an expression for the coordinates of 𝐶 .

  • A 𝑥 + 𝑘 𝑘 + 𝑚 ( 𝑥 − 𝑥 ) , 𝑦 + 𝑘 𝑘 + 𝑚 ( 𝑦 − 𝑦 )
  • B 𝑥 + 𝑘 𝑘 + 𝑚 ( 𝑥 − 𝑥 ) , 𝑦 + 𝑘 𝑘 + 𝑚 ( 𝑦 − 𝑦 )
  • C 𝑥 − 𝑘 𝑘 + 𝑚 ( 𝑥 − 𝑥 ) , 𝑦 − 𝑘 𝑘 + 𝑚 ( 𝑦 − 𝑦 )
  • D 𝑥 + 𝑚 𝑘 + 𝑚 ( 𝑥 − 𝑥 ) , 𝑦 + 𝑚 𝑘 + 𝑚 ( 𝑦 − 𝑦 )
  • E 𝑥 + 𝑘 𝑘 + 𝑚 ( 𝑥 − 𝑥 ) , 𝑦 + 𝑘 𝑘 + 𝑚 ( 𝑦 − 𝑦 )

Q19:

A triangle has vertices at the points 𝐴 ( 4 , 1 ) , 𝐵 ( 6 , 2 ) , and 𝐶 ( 9 , 0 ) .

Work out the lengths of the sides of the triangle. Give your answers as surds in their simplest form.

  • A 𝐴 𝐵 = √ 5 , 𝐴 𝐶 = 5 √ 2 , and 𝐵 𝐶 = √ 5
  • B 𝐴 𝐵 = 5 , 𝐴 𝐶 = 2 6 , and 𝐵 𝐶 = 1 3
  • C 𝐴 𝐵 = √ 5 , 𝐴 𝐶 = √ 1 6 , and 𝐵 𝐶 = √ 1 1
  • D 𝐴 𝐵 = √ 5 , 𝐴 𝐶 = √ 2 6 , and 𝐵 𝐶 = √ 1 3
  • E 𝐴 𝐵 = 5 , 𝐴 𝐶 = 1 6 , and 𝐵 𝐶 = 1 1

Is the triangle a right scalene triangle?

  • AYes
  • BIt is impossible to tell.
  • CNo

Q20:

The points 𝐴 , 𝐵 , 𝐶 , and 𝐷 have coordinates ( 3 , 3 ) , ( 9 , 5 ) , ( − 2 , 8 ) , and ( 3 , − 1 ) . Which of the line segments 𝐴 𝐵 and 𝐶 𝐷 has the greatest length?

  • A 𝐶 𝐷
  • B 𝐴 𝐵

Q21:

Find the distance between the two points in the figure below, giving your answer in radical form if necessary.

  • A √ 3 5
  • B7
  • C 5 √ 5
  • D √ 3 7
  • E 5 √ 3

Q22:

Find the distance between the two points in the figure below, giving your answer in radical form if necessary.

  • A 3 √ 3
  • B9
  • C53
  • D √ 4 5
  • E √ 5 3

Q23:

Let us consider 𝐷 ( − 1 2 , 5 ) in a coordinate system of origin 𝑂 ( 0 , 0 ) . Using the Pythagorean theorem, find the distance between 𝑂 and 𝐷 .

Q24:

Let us consider 𝐹 ( 1 3 , − 7 ) and 𝐺 ( − 1 1 , 3 ) in a coordinate system of origin 𝑂 ( 0 , 0 ) . Using the Pythagorean theorem, find the distance between 𝐹 and 𝐺 .

Q25:

Let us consider 𝐸 ( − 1 5 , − 8 ) in a coordinate system of origin 𝑂 ( 0 , 0 ) . Using the Pythagorean theorem, find the distance between 𝑂 and 𝐸 .

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