Worksheet: Coordinates of a Point in a Cartesian Plane

In this worksheet, we will practice finding the coordinates of a point that divides a line segment on the coordinate plane with a ratio.

Q1:

If the coordinates of 𝐴 and 𝐡 are ( 5 , 5 ) and ( βˆ’ 1 , βˆ’ 4 ) respectively, find the coordinates of the point 𝐢 that divides οƒ  𝐴 𝐡 internally by the ratio 2 ∢ 1 .

  • A ( βˆ’ 1 , 1 )
  • B ( 3 , 2 )
  • C ( βˆ’ 1 , βˆ’ 1 )
  • D ( 1 , βˆ’ 1 )

Q2:

The coordinates of points 𝐴 and 𝐡 are ( 4 , 4 ) and ( 1 , βˆ’ 2 ) respectively. Given that βƒ–     βƒ— 𝐴 𝐡 intersects the π‘₯ -axis at 𝐢 and the 𝑦 -axis at 𝐷 , find the ratio by which οƒ  𝐴 𝐡 is divided by points 𝐢 and 𝐷 respectively showing the type of division in each case.

  • A internal division by ratio 1 ∢ 4 , external division by ratio 1 ∢ 2
  • B internal division by ratio 4 ∢ 1 , external division by ratio 2 ∢ 1
  • C internal division by ratio 1 ∢ 2 , external division by ratio 1 ∢ 4
  • D internal division by ratio 2 ∢ 1 , external division by ratio 4 ∢ 1

Q3:

The coordinates of 𝐴 and 𝐡 are ( 1 , 9 ) and ( 9 , 9 ) respectively. Determine the coordinates of the points that divide 𝐴 𝐡 into four equal parts.

  • A ( 9 , 5 ) , ( 9 , 7 ) , ( βˆ’ 4 , 2 )
  • B ( 5 , 9 ) , ( 7 , 9 ) , ( βˆ’ 2 , 0 )
  • C ( 5 , 9 ) , ( 7 , 9 ) , ( 7 , 5 )
  • D ( 5 , 9 ) , ( 7 , 9 ) , ( 3 , 9 )

Q4:

A bus is travelling from city 𝐴 ( 1 0 , βˆ’ 1 0 ) to city 𝐡 ( βˆ’ 8 , 8 ) . Its first stop is at 𝐢 , which is halfway between the cities. Its second stop is at 𝐷 , which is two-thirds of the way from 𝐴 to 𝐡 . What are the coordinates of 𝐢 and 𝐷 ?

  • A ( 1 , βˆ’ 1 ) , ( 4 , βˆ’ 4 )
  • B ( 2 , βˆ’ 2 ) , ( βˆ’ 2 , 2 )
  • C ( 0 , 0 ) , ( βˆ’ 3 , 3 )
  • D ( 1 , βˆ’ 1 ) , ( βˆ’ 2 , 2 )

Q5:

Given 𝐴 ( βˆ’ 5 , 9 ) and 𝐡 ( 7 , βˆ’ 3 ) , what are the points 𝐢 and 𝐷 that divide 𝐴 𝐡 into three parts of equal length?

  • A ο€Ό 2 3 , 2  , ο€Ό 2 3 6 , βˆ’ 1 2 
  • B ( 1 , 3 ) , ( 1 , 3 )
  • C ο€Ό 4 3 , 4  , ο€Ό 2 3 , 2 
  • D ( βˆ’ 1 , 5 ) , ( 3 , 1 )

Q6:

Consider 𝐴 ( βˆ’ 1 , βˆ’ 2 ) and 𝐡 ( βˆ’ 7 , 7 ) . Find the coordinates of 𝐢 , given that 𝐢 is on the ray  𝐴 𝐡 but NOT on the segment 𝐴 𝐡 and 𝐴 𝐢 = 2 𝐢 𝐡 .

  • A ( 5 , βˆ’ 1 1 )
  • B ( βˆ’ 5 , 4 )
  • C ( βˆ’ 3 , 1 )
  • D ( βˆ’ 1 3 , 1 6 )

Q7:

Consider points 𝐴 ( 2 , 3 ) and 𝐡 ( βˆ’ 4 , βˆ’ 3 ) . Find the coordinates of 𝐢 , given that 𝐢 is on the ray  𝐡 𝐴 but NOT on the segment 𝐴 𝐡 and 𝐴 𝐢 = 2 𝐴 𝐡 .

  • A ( 8 , 9 )
  • B ( βˆ’ 2 , 3 )
  • C ( 0 , 1 )
  • D ( 1 4 , 1 5 )

Q8:

Given 𝐴 ( 6 , βˆ’ 6 ) and 𝐡 ( βˆ’ 7 , βˆ’ 1 ) , find the coordinates of 𝐢 on βƒ–     βƒ— 𝐴 𝐡 for which 2 𝐴 𝐢 = 9 𝐢 𝐡 .

  • A ο€Ό βˆ’ 5 1 7 , βˆ’ 3  , ( βˆ’ 7 5 , 3 )
  • B ( βˆ’ 5 1 , βˆ’ 2 1 ) , ( βˆ’ 7 5 , 3 )
  • C ( 5 1 , 2 1 ) , ( βˆ’ 7 5 , 3 )
  • D ο€Ό βˆ’ 5 1 1 1 , βˆ’ 2 1 1 1  , ο€Ό βˆ’ 7 5 7 , 3 7 

Q9:

The coordinates of the points 𝐴 and 𝐡 are ( βˆ’ 3 , 4 ) and ( βˆ’ 4 , βˆ’ 2 ) respectively. Determine the coordinates of the point 𝐢 , given that it divides  𝐴 𝐡 externally in the ratio 2 ∢ 1 .

  • A ( βˆ’ 8 , βˆ’ 5 )
  • B ( βˆ’ 2 , 1 0 )
  • C ( 5 , βˆ’ 8 )
  • D ( βˆ’ 5 , βˆ’ 8 )

Q10:

Suppose 𝐴 ( 1 , 3 ) and another point 𝐡 , and that 𝐢 ( 5 , 1 ) divides 𝐴 𝐡 internally in the ratio 2 ∢ 3 . What are the coordinates of 𝐡 ?

  • A ( 1 4 , 7 )
  • B ( 2 2 , βˆ’ 4 )
  • C ( 2 8 , 1 4 )
  • D ( 1 1 , βˆ’ 2 )

Q11:

Line segment 𝐴 𝐷 is a median in β–³ 𝐴 𝐡 𝐢 , where 𝐴 = ( 8 , βˆ’ 7 ) and 𝐷 = ( 2 , βˆ’ 1 ) . Find the point of intersection of the medians of the triangle 𝐴 𝐡 𝐢 .

  • A ( 1 2 , βˆ’ 9 )
  • B ( 6 , βˆ’ 5 )
  • C ( 1 8 , βˆ’ 1 5 )
  • D ( 4 , βˆ’ 3 )

Q12:

Find the ratio by which the -axis divides the line segment , joining points and , showing the type of division, and determine the coordinates of the point of division.

  • A internally,
  • B internally,
  • C externally,
  • D externally,

Q13:

Given that the coordinates of the points 𝐴 and 𝐡 are ( 9 , 6 ) and ( βˆ’ 1 , 6 ) respectively, determine, in vector form, the coordinates of the point 𝐢 , which divides οƒ  𝐴 𝐡 internally in the ratio 4 ∢ 1 .

  • A ( 6 , 1 )
  • B ( 7 , 6 )
  • C ( βˆ’ 1 , 6 )
  • D ( 1 , 6 )

Q14:

Given points 𝐴 ( βˆ’ 2 , βˆ’ 6 ) and 𝐡 ( βˆ’ 7 , 4 ) , find the ratio by which the π‘₯ -axis divided line segment 𝐴 𝐡 , together with the type of division. Determine the coordinates of the point of intersection.

  • A 2 ∢ 7 externally, ( βˆ’ 1 0 , 0 )
  • B 3 ∢ 2 externally, ( βˆ’ 5 , 0 )
  • C 2 ∢ 7 internally, ( βˆ’ 1 0 , 0 )
  • D 3 ∢ 2 internally, ( βˆ’ 5 , 0 )

Q15:

If the coordinates of the points 𝐴 and 𝐡 are ( 9 , βˆ’ 1 ) and ( 2 , βˆ’ 1 ) , respectively, find the ratio by which the point 𝐢 ( 7 , 𝑦 ) divides 𝐴 𝐡 stating whether it is divided internally or externally, then determine the value of 𝑦 .

  • A 5 ∢ 2 internally, 𝑦 = 1
  • B 2 ∢ 5 externally, 𝑦 = βˆ’ 1
  • C 5 ∢ 2 externally, 𝑦 = 1
  • D 2 ∢ 5 internally, 𝑦 = βˆ’ 1

Q16:

If 𝐴 ( 3 , βˆ’ 2 ) and 𝐡 ( βˆ’ 2 , 4 ) , find in vector form the coordinates of point 𝐢 which divides οƒ  𝐴 𝐡 externally in the ratio 4 ∢ 3 .

  • A ( 2 2 , βˆ’ 1 7 )
  • B ( 1 8 , βˆ’ 2 0 )
  • C ( 1 7 , 2 2 )
  • D ( βˆ’ 1 7 , 2 2 )

Q17:

If 𝐴 ( βˆ’ 1 5 , βˆ’ 7 ) , 𝐡 ( 7 , 2 ) , 𝐢 ( 4 , βˆ’ 1 7 ) , 𝐷 ( 1 3 , βˆ’ 2 ) , 𝐸 is the midpoint of 𝐴 𝐡 , and 𝑀 divides 𝐢 𝐷 externally by the ratio 7 ∢ 4 , find the length of 𝐸 𝑀 to the nearest hundredth considering a length unit = 1 c m .

Q18:

If the coordinates of 𝐴 and 𝐡 are ( 3 , 1 ) and ( βˆ’ 7 , 1 ) respectively, find the coordinates of the point 𝐢 that divides οƒ  𝐴 𝐡 internally by the ratio 2 ∢ 3 .

  • A ( 1 , βˆ’ 1 )
  • B ( βˆ’ 3 , 1 )
  • C ( 1 , 1 )
  • D ( βˆ’ 1 , 1 )

Q19:

The coordinates of the points 𝐴 and 𝐡 are ( 2 , 2 ) and ( 5 , 1 ) respectively. Determine the coordinates of the point 𝐢 , given that it divides  𝐴 𝐡 externally in the ratio 4 ∢ 3 .

  • A ( βˆ’ 2 , 1 4 )
  • B ( βˆ’ 7 , 5 )
  • C ( βˆ’ 1 4 , βˆ’ 2 )
  • D ( 1 4 , βˆ’ 2 )

Q20:

The coordinates of points 𝐴 and 𝐡 are ( 6 , 6 ) and ( 1 , βˆ’ 4 ) respectively. Given that βƒ–     βƒ— 𝐴 𝐡 intersects the π‘₯ -axis at 𝐢 and the 𝑦 -axis at 𝐷 , find the ratio by which οƒ  𝐴 𝐡 is divided by points 𝐢 and 𝐷 respectively showing the type of division in each case.

  • A internal division by ratio 1 ∢ 6 , external division by ratio 2 ∢ 3
  • B internal division by ratio 6 ∢ 1 , external division by ratio 3 ∢ 2
  • C internal division by ratio 2 ∢ 3 , external division by ratio 1 ∢ 6
  • D internal division by ratio 3 ∢ 2 , external division by ratio 6 ∢ 1

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