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Lesson: Distance between Two Points on a Coordinate Plane

Video

14:22

Sample Question Videos

Worksheet • 25 Questions • 3 Videos

Q1:

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

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

Q2:

Find the distance between the points 𝐴 and 𝐵 .

  • A 5 8 length units
  • B16 length units
  • C 1 0 length units
  • D 2 1 3 length units
  • E4 length units

Q3:

Find the distance between the points 𝐴 and 𝐵 .

  • A 7 4 length units
  • B21 length units
  • C 2 3 length units
  • D 4 2 length units
  • E 3 6 length units

Q4:

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 = 4
  • C 𝑎 = 4 or 𝑎 = 5
  • D 𝑎 = 5 or 𝑎 = 5
  • E 𝑎 = 4 or 𝑎 = 5

Q5:

If the distance between the two points ( 𝑎 , 6 ) and ( 2 𝑎 4 , 1 0 ) is 5, find all possible values of 𝑎 .

  • A 𝑎 = 1 or 𝑎 = 7
  • B 𝑎 = 1 or 𝑎 = 1
  • C 𝑎 = 1 or 𝑎 = 7
  • D 𝑎 = 7 or 𝑎 = 7
  • E 𝑎 = 1 or 𝑎 = 7

Q6:

If the distance between the two points ( 𝑎 , 9 ) and ( 2 𝑎 + 2 , 1 ) is 10, find all possible values of 𝑎 .

  • A 𝑎 = 8 or 𝑎 = 4
  • B 𝑎 = 4 or 𝑎 = 4
  • C 𝑎 = 8 or 𝑎 = 4
  • D 𝑎 = 8 or 𝑎 = 8
  • E 𝑎 = 8 or 𝑎 = 4

Q7:

How far is the butcher from the garage?

Q8:

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.

Q9:

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.

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:

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 𝐴 𝐵
  • B 𝐶 𝐷

Q12:

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

  • A 5 3 length units
  • B13 length units
  • C3 length units
  • D 5 1 length units
  • E 1 1 length units

Q13:

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

  • A 8 5 length units
  • B14 length units
  • C 1 3 length units
  • D 5 5 length units
  • E 4 3 length units

Q14:

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

  • A 3 7
  • B 5 3
  • C7
  • D 5 5
  • E 3 5

Q15:

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

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

Q16:

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

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

Q17:

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

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
  • E 𝑥 + 𝑥 2

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

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

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 + 𝑚 𝑘 + 𝑚 ( 𝑥 𝑥 ) , 𝑦 + 𝑚 𝑘 + 𝑚 ( 𝑦 𝑦 )
  • D 𝑥 𝑘 𝑘 + 𝑚 ( 𝑥 𝑥 ) , 𝑦 𝑘 𝑘 + 𝑚 ( 𝑦 𝑦 )
  • E 𝑥 + 𝑘 𝑘 + 𝑚 ( 𝑥 𝑥 ) , 𝑦 + 𝑘 𝑘 + 𝑚 ( 𝑦 𝑦 )

Q18:

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 rescue boat
  • BThe coast guard craft

Q19:

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

  • A 1 6 + 4 1 3
  • B 1 6 + 2 1 0
  • C 1 6 + 2 5
  • D18

Q20:

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
  • B6
  • C 2
  • D8
  • E2

Q21:

A triangle has vertices at the points 𝐴 ( 4 , 1 ) , 𝐵 ( 6 , 2 ) , 𝐶 ( 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 𝐵 = 5 , 𝐴 𝐶 = 2 6 , 𝐵 𝐶 = 1 3 a n d
  • B 𝐴 𝐵 = 5 , 𝐴 𝐶 = 1 6 , 𝐵 𝐶 = 1 1 a n d
  • C 𝐴 𝐵 = 5 , 𝐴 𝐶 = 2 6 , 𝐵 𝐶 = 1 3 a n d
  • D 𝐴 𝐵 = 5 , 𝐴 𝐶 = 1 6 , 𝐵 𝐶 = 1 1 a n d
  • E 𝐴 𝐵 = 5 , 𝐴 𝐶 = 5 2 , 𝐵 𝐶 = 5 a n d

Is the triangle a right scalene triangle?

  • Ano
  • Byes
  • CIt is impossible to tell.

Q22:

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

Q23:

If 𝐴 ( 6 , 𝑥 ) and 𝐵 ( 1 2 , 9 ) , where 𝐴 𝐵 = 2 1 4 5 length units, find all the possible values of 𝑥 .

  • A 𝑥 = 2 5 or 𝑥 = 7
  • B 𝑥 = 2 5 or 𝑥 = 7
  • C 𝑥 = 2 5 or 𝑥 = 7
  • D 𝑥 = 2 5 or 𝑥 = 7

Q24:

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

Q25:

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.

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