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Lesson: Distances on the Coordinate Plane

In this lesson, we will learn how to find the distance between two points and use it to determine the areas of certain shapes.

Sample Question Videos

04:19

Worksheet • 25 Questions • 1 Video

Q1:

Quadrilateral 𝑃 𝑄 𝑅 𝑆 has vertices 𝑃 ( βˆ’ 1 , 6 ) , 𝑄 ( 5 , 6 ) , 𝑅 ( 5 , 3 ) , and 𝑆 ( βˆ’ 1 , 3 ) . Find the length of 𝑄 𝑆 .

  • A 3 √ 5
  • B √ 6
  • C √ 9 7
  • D 5 √ 2
  • E3

Q2:

What is the distance between the points ( βˆ’ 8 , 2 ) and ( βˆ’ 4 , 2 ) ?

Q3:

A teacher asked his students to find the distance between the points and . Fares said that the distance is 4, while Karim said that the distance is 6. Which of them answered correctly?

  • AFares
  • BKarim

Q4:

Find the lengths of 𝐴 𝐡 and 𝐷 𝐢 , where the coordinates of points 𝐴 , 𝐡 , 𝐢 , and 𝐷 are ( βˆ’ 2 , 3 ) , ( 5 , 3 ) , ( βˆ’ 2 , βˆ’ 4 ) , and ( βˆ’ 2 , βˆ’ 5 ) , respectively, considering that a length unit is equal to 1 cm.

  • A 𝐴 𝐡 = 7 c m , 𝐷 𝐢 = 1 c m
  • B 𝐴 𝐡 = 5 c m , 𝐷 𝐢 = 2 c m
  • C 𝐴 𝐡 = 7 c m , 𝐷 𝐢 = 9 c m
  • D 𝐴 𝐡 = 2 c m , 𝐷 𝐢 = 3 c m

Q5:

Ramy and Fares are two friends who go to the same school. Ramy’s room is located at , Fares’s room at , and the band room at .

How far is Ramy’s room from Fares’s?

What is the distance between Fares’s room and the band room?

Q6:

In a treasure hunt, a sealed envelope was given to each team. The envelope contained map drawn on a coordinate grid and the following information: the campsite is at the origin point, the campfire at ( 5 , 4 ) , the river at ( 5 , βˆ’ 5 ) , the tent at ( βˆ’ 2 , βˆ’ 5 ) , and the big tree at ( βˆ’ 2 , 4 ) .

What is the distance between the campfire and the big tree?

What is the distance between the tent and the big tree?

What is the distance between the campfire and the river?

Q7:

Suppose that in the given figure, 𝐡 ( 4 3 . 5 , βˆ’ 4 3 . 5 ) and 𝐴 is on the π‘₯ -axis. What is the length of 𝐴 𝐡 ?

Q8:

Find the length of the base of Triangle B.

Q9:

Shady graphed the points 𝐴 ( 5 , 2 ) and 𝐷 ( βˆ’ 1 , 2 ) . He said that the distance between these points is 4. Do you agree or disagree?

  • Aagree
  • Bdisagree

Q10:

What is the length of the line in the figure below? Give your answer correct to one decimal place if necessary.

Q11:

Find the distance between π‘ˆ ( βˆ’ 7 , βˆ’ 3 ) and 𝑉 ( 3 , 2 ) , giving your answer correct to one decimal place if necessary.

Q12:

Find the distance between the two points in the figure below. Give your answer correct to one decimal place if necessary.

Q13:

Find the distance between 𝑆 ( 2 , βˆ’ 8 ) and 𝑇 ( βˆ’ 3 , 5 ) , giving your answer correct to one decimal place if necessary.

Q14:

Find the distance between point 𝐴 , located at ( 9 , βˆ’ 3 ) , and point 𝐡 , located at ( 4 , βˆ’ 3 ) .

Q15:

If 𝐴 ( π‘₯ , βˆ’ 2 ) , 𝐡 ( 0 , 5 ) , 𝐢 ( 7 , 1 2 ) , and 𝐴 𝐡 = 𝐡 𝐢 , find all the possible values of π‘₯ .

  • A π‘₯ = βˆ’ 7 or π‘₯ = 7
  • B π‘₯ = 0 or π‘₯ = 5
  • C π‘₯ = βˆ’ 1 2 or π‘₯ = βˆ’ 7
  • D π‘₯ = 7 or π‘₯ = 1 2

Q16:

Find the length of 𝐴 𝐡 .

Q17:

Do the two points ( βˆ’ 5 , 5 ) and ( βˆ’ 1 , 1 ) lie on the same side of the line 3 π‘₯ βˆ’ 2 𝑦 βˆ’ 4 = 0 ?

  • Ano
  • Byes

Q18:

Use the graph below to determine the coordinates of the points 𝐴 , 𝐡 , and 𝐢 , then find the area of the resulting figure from connecting the points.

  • A 𝐴 ( βˆ’ 2 , 0 ) , 𝐡 ( βˆ’ 2 , 5 ) , 𝐢 ( 2 , 5 ) , 10 area units
  • B 𝐴 ( βˆ’ 2 , 0 ) , 𝐡 ( βˆ’ 2 , 5 ) , 𝐢 ( 2 , 5 ) , 20 area units
  • C 𝐢 ( βˆ’ 2 , 0 ) , 𝐡 ( βˆ’ 2 , 5 ) , 𝐴 ( 2 , 5 ) , 20 area units
  • D 𝐡 ( βˆ’ 2 , 0 ) , 𝐢 ( βˆ’ 2 , 5 ) , 𝐴 ( 2 , 5 ) , 10 area units

Q19:

Use the graph below to determine the points 𝐴 , 𝐡 , 𝐢 , and 𝐷 , and find the area of the shape that results from connecting them.

  • A 𝐴 ( βˆ’ 4 , 6 ) , 𝐡 ( βˆ’ 4 , βˆ’ 6 ) , 𝐢 ( 4 , βˆ’ 6 ) , 𝐷 ( 4 , 6 ) , 96
  • B 𝐴 ( βˆ’ 4 , 6 ) , 𝐡 ( βˆ’ 4 , βˆ’ 6 ) , 𝐢 ( 4 , βˆ’ 6 ) , 𝐷 ( 4 , 6 ) , 48
  • C 𝐡 ( βˆ’ 4 , 6 ) , 𝐢 ( βˆ’ 4 , βˆ’ 6 ) , 𝐴 ( 4 , βˆ’ 6 ) , 𝐷 ( 4 , 6 ) , 48
  • D 𝐡 ( βˆ’ 4 , 6 ) , 𝐢 ( βˆ’ 4 , βˆ’ 6 ) , 𝐴 ( 4 , βˆ’ 6 ) , 𝐷 ( 4 , 6 ) , 96

Q20:

What is the distance between the point ( 5 , 1 9 ) and the 𝑦 -axis?

Q21:

Given that the coordinates of the points 𝐴 and 𝐡 are ( 1 3 , 8 ) and ( 1 4 , 8 ) respectively, and one unit length = 1 c m , find the length of 𝐴 𝐡 .

Q22:

What is the length of the line in the figure below? Give your answer correct to one decimal place if necessary.

Q23:

Given points 𝐢 ( 1 6 , 2 0 ) and 𝐷 ( 1 6 , 1 0 ) , calculate the distance between the two points, 𝐢 and 𝐷 , considering that a length unit = 1 c m .

Q24:

Given that one length unit equals 1 cm, find the perimeter of 𝐿 𝑀 𝑁 𝐻 , where the coordinates of points 𝐿 , 𝑀 , 𝑁 , and 𝐻 are ( βˆ’ 7 , βˆ’ 3 ) , ( 2 , βˆ’ 3 ) , ( 2 , 9 ) , and ( βˆ’ 7 , 9 ) , respectively.

Q25:

Given that the coordinates of 𝑀 and 𝑁 are ( 6 , 5 ) and ( 6 , 1 1 ) , respectively, determine the length of 𝑀 𝑁 .

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