Worksheet: Distance on the Coordinate Plane: Pythagorean Formula

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

Q2:

Find the distance between the points 𝐴 and 𝐵.

  • A4 length units
  • B16 length units
  • C 5 8 length units
  • D 2 1 3 length units
  • E 1 0 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 𝐴𝐵.

  • 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.

  • A8
  • B2
  • C 2
  • D6
  • E 2 2

Q4:

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

Q5:

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

  • A 𝑎 = 5 or 𝑎=5
  • B 𝑎 = 4 or 𝑎=5
  • C 𝑎 = 4 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 52 from the origin?

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

Q7:

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

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

Q8:

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

  • A 5 1 length units
  • B 5 3 length units
  • C13 length units
  • D 1 1 length units
  • E3 length units

Q9:

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

Q10:

If 𝐴(7,𝑥) and 𝐵(9,14), where 𝐴𝐵=417length units, find all the possible values of 𝑥.

  • A 𝑥 = 1 0 or 𝑥=18
  • B 𝑥 = 1 0 or 𝑥=18
  • C 𝑥 = 1 0 or 𝑥=18
  • D 𝑥 = 1 0 or 𝑥=18

Q11:

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

Q12:

William 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.

Q13:

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

Q14:

A small craft in Lake Ontario sends out a distress signal from the coordinates (49,64). One rescue boat is at the coordinates (60,82), while a Coast Guard boat one is at the coordinates (58,47). Assuming both boat travel at the same rate, which one will get to the distressed boat first?

  • AThe rescue boat
  • BThe coast guard craft

Q15:

On a map, coordinates are expressed in miles. The coordinates for San Francisco are (53,17) and those for Sacramento are (123,78). Find the distance between the cities to the nearest mile.

Q16:

On a map, each coordinate represents a distance in miles from some fixed point. San Jose’s coordinates are (76, 12), while those for San Francisco are (53, 17). Find the distance between San Jose and San Francisco to the nearest mile.

Q17:

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 = 2 6 , and 𝐵𝐶=13
  • B 𝐴 𝐵 = 5 , 𝐴 𝐶 = 1 6 , and 𝐵𝐶=11
  • C 𝐴 𝐵 = 5 , 𝐴 𝐶 = 5 2 , and 𝐵𝐶=5
  • D 𝐴 𝐵 = 5 , 𝐴 𝐶 = 1 6 , and 𝐵𝐶=11
  • E 𝐴 𝐵 = 5 , 𝐴 𝐶 = 2 6 , and 𝐵𝐶=13

Is the triangle a right scalene triangle?

  • AYes
  • BNo
  • CIt is impossible to tell.

Q18:

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

  • A 𝐴 𝐵
  • B 𝐶 𝐷

Q19:

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

  • A 3 5
  • B 5 5
  • C 3 7
  • D7
  • E 5 3

Q20:

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

  • A53
  • B9
  • C 4 5
  • D 5 3
  • E 3 3

Q21:

Let us consider 𝐷(12,5) in a coordinate system of origin 𝑂(0,0). Using the Pythagorean theorem, find the distance between 𝑂 and 𝐷.

Q22:

Let us consider 𝐹(13,7) and 𝐺(11,3) in a coordinate system of origin 𝑂(0,0). Using the Pythagorean theorem, find the distance between 𝐹 and 𝐺.

Q23:

Let us consider 𝐸(15,8) in a coordinate system of origin 𝑂(0,0). Using the Pythagorean theorem, find the distance between 𝑂 and 𝐸.

Q24:

Let us consider 𝐹(8,6) in a coordinate system of origin 𝑂(0,0). Using the Pythagorean theorem, find the distance between 𝑂 and 𝐹.

Q25:

Let us consider 𝐷(14,9) and 𝐸(10,2) in a coordinate system of origin 𝑂(0,0). Using the Pythagorean theorem, find the distance between 𝐷 and 𝐸.

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