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Worksheet: Finding the Distance between Two Points on a Coordinate Plane

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

Q1:

Consider the two points 𝐴 ( π‘₯ , 𝑦 ) 1 1 and 𝐡 ( π‘₯ , 𝑦 ) 2 2 .

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

  • A √ ( π‘₯ βˆ’ π‘₯ ) + ( 𝑦 βˆ’ 𝑦 ) 1 2 1 2
  • B  ( π‘₯ + π‘₯ ) βˆ’ ( 𝑦 + 𝑦 ) 1 2 2 1 2 2
  • C ( π‘₯ βˆ’ π‘₯ ) + ( 𝑦 βˆ’ 𝑦 ) 1 2 2 1 2 2
  • D  ( π‘₯ βˆ’ π‘₯ ) + ( 𝑦 βˆ’ 𝑦 ) 1 2 2 1 2 2
  • E ( π‘₯ + π‘₯ ) βˆ’ ( 𝑦 + 𝑦 ) 1 2 2 1 2 2

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

Q2:

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

Q3:

Find the distance between the point ( βˆ’ 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

Q4:

Find the distance between the points 𝐴 and 𝐡 .

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

Q5:

Find the distance between the points 𝐴 and 𝐡 .

  • A 3 √ 6 length units
  • B 2 √ 3 length units
  • C 4 √ 2 length units
  • D √ 7 4 length units
  • E21 length units

Q6:

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

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

Q7:

Find the distance between the points ( 2 , 3 ) and ( βˆ’ 4 , βˆ’ 4 ) .

  • A √ 4 3 length units
  • B √ 1 3 length units
  • C √ 5 5 length units
  • D √ 8 5 length units
  • E14 length units

Q8:

The distance between ( π‘Ž , 5 ) and ( 1 , 1 ) is 5. What are the possible values of π‘Ž ?

  • A π‘Ž = 2 or βˆ’ 4
  • B π‘Ž = 2 or 4
  • C π‘Ž = βˆ’ 2 or βˆ’ 4
  • D π‘Ž = βˆ’ 2 or 4

Q9:

If 𝐴 ( 6 , π‘₯ ) and 𝐡 ( βˆ’ 1 2 , βˆ’ 9 ) , where 𝐴 𝐡 = 2 √ 1 4 5 length units, find all the possible values of π‘₯ .

  • A π‘₯ = βˆ’ 2 5 or π‘₯ = βˆ’ 7
  • B π‘₯ = 2 5 or π‘₯ = βˆ’ 7
  • C π‘₯ = 2 5 or π‘₯ = 7
  • D π‘₯ = βˆ’ 2 5 or π‘₯ = 7

Q10:

If the distance between the two points ( π‘Ž , 0 ) and ( βˆ’ π‘Ž + 1 , 0 ) is 9, find all possible values of π‘Ž .

  • A π‘Ž = 4 or π‘Ž = βˆ’ 5
  • B π‘Ž = 4 or π‘Ž = 5
  • C π‘Ž = 5 or π‘Ž = βˆ’ 5
  • D π‘Ž = βˆ’ 4 or π‘Ž = 5
  • E π‘Ž = βˆ’ 4 or π‘Ž = 4

Q11:

If the distance between the two points ( π‘Ž , βˆ’ 6 ) and ( 2 π‘Ž βˆ’ 4 , βˆ’ 1 0 ) is 5, find all possible values of π‘Ž .

  • A π‘Ž = βˆ’ 1 or π‘Ž = βˆ’ 7
  • B π‘Ž = βˆ’ 1 or π‘Ž = 7
  • C π‘Ž = 7 or π‘Ž = βˆ’ 7
  • D π‘Ž = 1 or π‘Ž = 7
  • E π‘Ž = 1 or π‘Ž = βˆ’ 1

Q12:

If the distance between the two points ( π‘Ž , βˆ’ 9 ) and ( 2 π‘Ž + 2 , βˆ’ 1 ) is 10, find all possible values of π‘Ž .

  • A π‘Ž = 8 or π‘Ž = βˆ’ 4
  • B π‘Ž = 8 or π‘Ž = 4
  • C π‘Ž = βˆ’ 8 or π‘Ž = 8
  • D π‘Ž = βˆ’ 8 or π‘Ž = 4
  • E π‘Ž = 4 or π‘Ž = βˆ’ 4

Q13:

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

  • A ( 5 , 0 )
  • B ( 0 , 5 )
  • C ο€» 5 √ 2 , 5 √ 2 
  • D ( 5 , 5 )

Q14:

Given 𝐴 ( 4 , 5 ) , 𝐡 ( 5 , 5 ) , and 𝐢 ( βˆ’ 4 , βˆ’ 7 ) , what is the perimeter of β–³ 𝐴 𝐡 𝐢 ?

  • A18
  • B ο€» 1 6 + 2 √ 1 0 
  • C ο€» 1 6 + 2 √ 5 
  • D ο€» 1 6 + 4 √ 1 3 

Q15:

How far is the butcher from the garage?

Q16:

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

Q17:

William is making a map of his local area measured in yards. The coffee shop is at ( βˆ’ 5 , βˆ’ 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.

Q18:

The point 𝑃 is located on the line segment between point 𝐴 ( βˆ’ 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 )

Q19:

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

Q20:

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.

Q21:

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

Q22:

Points 𝐴 and 𝐡 have coordinates 𝐴 ( π‘₯ , 𝑦 ) 1 1 and 𝐡 ( π‘₯ , 𝑦 ) 2 2 respectively. Point 𝐢 lies on 𝐴 𝐡 such that 𝐴 𝐢 and 𝐡 𝐢 are in the ratio of π‘˜ ∢ π‘š .

Write the quotient of the length of 𝐴 𝐢 by the length of 𝐴 𝐡 in terms of π‘˜ and π‘š .

  • A π‘˜ + π‘š π‘˜
  • B π‘˜ π‘š
  • C π‘š π‘˜
  • D π‘˜ π‘˜ + π‘š
  • E π‘š π‘˜ + π‘š

Find an expression for the difference between the π‘₯ coordinates of 𝐴 and 𝐡 .

  • A π‘₯ βˆ’ π‘₯ 2 1
  • B π‘₯ π‘₯ 2 1
  • C π‘₯ + π‘₯ 2 2 1
  • D π‘₯ βˆ’ π‘₯ 2 2 1
  • E π‘₯ π‘₯ 1 2

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

  • A 𝑦 𝑦 1 2
  • B 𝑦 βˆ’ 𝑦 2 2 1
  • C 𝑦 βˆ’ 𝑦 2 1
  • D 𝑦 + 𝑦 2 2 1
  • E 𝑦 𝑦 2 1

Find an expression for the coordinates of 𝐢 .

  • A ο€½ π‘₯ + π‘˜ π‘˜ + π‘š ( π‘₯ βˆ’ π‘₯ ) , 𝑦 + π‘˜ π‘˜ + π‘š ( 𝑦 βˆ’ 𝑦 )  1 2 1 1 2 1
  • B ο€½ π‘₯ + π‘˜ π‘˜ + π‘š ( π‘₯ βˆ’ π‘₯ ) , 𝑦 + π‘˜ π‘˜ + π‘š ( 𝑦 βˆ’ 𝑦 )  1 2 1 1 1 2
  • C ο€½ π‘₯ βˆ’ π‘˜ π‘˜ + π‘š ( π‘₯ βˆ’ π‘₯ ) , 𝑦 βˆ’ π‘˜ π‘˜ + π‘š ( 𝑦 βˆ’ 𝑦 )  1 2 1 1 2 1
  • D ο€Ό π‘₯ + π‘š π‘˜ + π‘š ( π‘₯ βˆ’ π‘₯ ) , 𝑦 + π‘š π‘˜ + π‘š ( 𝑦 βˆ’ 𝑦 )  1 2 1 1 2 1
  • E ο€½ π‘₯ + π‘˜ π‘˜ + π‘š ( π‘₯ βˆ’ π‘₯ ) , 𝑦 + π‘˜ π‘˜ + π‘š ( 𝑦 βˆ’ 𝑦 )  1 1 2 1 2 1

Q23:

A triangle has vertices at the points 𝐴 ( 4 , 1 ) , 𝐡 ( 6 , 2 ) , 𝐢 ( 9 , 0 ) a n d .

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

  • A 𝐴 𝐡 = √ 5 , 𝐴 𝐢 = 5 √ 2 , 𝐡 𝐢 = √ 5 a n d
  • B 𝐴 𝐡 = 5 , 𝐴 𝐢 = 2 6 , 𝐡 𝐢 = 1 3 a n d
  • C 𝐴 𝐡 = √ 5 , 𝐴 𝐢 = √ 1 6 , 𝐡 𝐢 = √ 1 1 a n d
  • D 𝐴 𝐡 = √ 5 , 𝐴 𝐢 = √ 2 6 , 𝐡 𝐢 = √ 1 3 a n d
  • E 𝐴 𝐡 = 5 , 𝐴 𝐢 = 1 6 , 𝐡 𝐢 = 1 1 a n d

Is the triangle a right scalene triangle?

  • Ayes
  • BIt is impossible to tell.
  • Cno

Q24:

The points 𝐴 , 𝐡 , 𝐢 , and 𝐷 have coordinates ( 3 , 3 ) , ( 9 , 5 ) , ( βˆ’ 2 , 8 ) , and ( 3 , βˆ’ 1 ) . Which of the line segments 𝐴 𝐡 and 𝐢 𝐷 has the greatest length?

  • A 𝐢 𝐷
  • B 𝐴 𝐡

Q25:

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

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