Worksheet: Distance on the Coordinate Plane: Pythagorean Formula

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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 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 5√2 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=4√17length 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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