Worksheet: Partitioning a Line Segment on the Coordinate 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 using the section formula.

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 𝐴𝐡 internally by the ratio 2∢1.

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

Q2:

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

  • Ainternal division by ratio 1∢4, external division by ratio 1∢2
  • Binternal division by ratio 4∢1, external division by ratio 2∢1
  • Cinternal division by ratio 1∢2, external division by ratio 1∢4
  • Dinternal 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 𝐴𝐡 into four equal parts.

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

Q4:

A bus is travelling from city 𝐴(10,βˆ’10) 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 ( 0 , 0 ) , ( βˆ’ 3 , 3 )
  • B ( 1 , βˆ’ 1 ) , ( 4 , βˆ’ 4 )
  • C ( 2 , βˆ’ 2 ) , ( βˆ’ 2 , 2 )
  • 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 𝐴𝐡 into three parts of equal length?

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

Q6:

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

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

Q7:

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

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

Q8:

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

  • A ( 5 1 , 2 1 ) , ( βˆ’ 7 5 , 3 )
  • B ο€Ό βˆ’ 5 1 7 , βˆ’ 3  , ( βˆ’ 7 5 , 3 )
  • C ( βˆ’ 5 1 , βˆ’ 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 𝐴𝐡 externally in the ratio 2∢1.

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

Q10:

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

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

Q11:

Line segment 𝐴𝐷 is a median in △𝐴𝐡𝐢, where 𝐴=(8,βˆ’7) and 𝐷=(2,βˆ’1). Find the point of intersection of the medians of the triangle 𝐴𝐡𝐢.

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

Q12:

Find the ratio by which the 𝑦-axis divides the line segment 𝐴𝐡, joining points 𝐴(βˆ’3,6) and 𝐡(βˆ’8,βˆ’4), showing the type of division, and determine the coordinates of the point of division.

  • A 3 ∢ 8 internally, (0,12)
  • B 3 ∢ 2 internally, (βˆ’6,0)
  • C 3 ∢ 2 externally , (βˆ’6,0)
  • D 3 ∢ 8 externally, (0,12)

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 𝐴𝐡 internally in the ratio 4∢1.

  • A ⟨ 7 , 6 ⟩
  • B ⟨ 6 , 1 ⟩
  • C ⟨ 1 , 6 ⟩
  • D ⟨ βˆ’ 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 𝐴𝐡, together with the type of division. Determine the coordinates of the point of intersection.

  • A 2 ∢ 7 externally, (βˆ’10,0)
  • B 3 ∢ 2 externally, (βˆ’5,0)
  • C 2 ∢ 7 internally, (βˆ’10,0)
  • 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 𝐴𝐡 stating whether it is divided internally or externally, then determine the value of 𝑦.

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

Q16:

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

  • A ⟨ 2 2 , βˆ’ 1 7 ⟩
  • B ⟨ 1 7 , 2 2 ⟩
  • C ⟨ 1 8 , βˆ’ 2 0 ⟩
  • D ⟨ βˆ’ 1 7 , 2 2 ⟩

Q17:

If 𝐴(βˆ’15,βˆ’7), 𝐡(7,2), 𝐢(4,βˆ’17), 𝐷(13,βˆ’2), 𝐸 is the midpoint of 𝐴𝐡, and 𝑀 divides 𝐢𝐷 externally by the ratio 7∢4, find the length of 𝐸𝑀 to the nearest hundredth considering a length unit =1cm.

Q18:

If the coordinates of 𝐴 and 𝐡 are (9,3) and (βˆ’3,βˆ’3) respectively, find the coordinates of the point 𝐢 that divides 𝐴𝐡 internally by the ratio 1∢2.

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

Q19:

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

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

Q20:

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

  • Ainternal division by ratio 1∢6, external division by ratio 2∢3
  • Binternal division by ratio 6∢1, external division by ratio 3∢2
  • Cinternal division by ratio 2∢3, external division by ratio 1∢6
  • Dinternal division by ratio 3∢2, external division by ratio 6∢1

Q21:

Two points 𝐴 and 𝐡 are at (1,2) and (4,βˆ’1) respectively. Point 𝐢 lies on the line segment 𝐴𝐡 such that the length of 𝐴𝐢 is 13 of 𝐴𝐡. Find the coordinates of 𝐢.

  • A 𝐢 = ( 1 , 2 )
  • B 𝐢 = ( 2 , 1 )
  • C 𝐢 = ( βˆ’ 1 , βˆ’ 1 )
  • D 𝐢 = ( 1 , 1 )
  • E 𝐢 = ( 1 , 0 )

Q22:

Two points 𝐴 and 𝐡 are at (5,βˆ’6) and (9,2) respectively. Point 𝐢 lies on the line segment 𝐴𝐡 such that the length of 𝐴𝐢 is 34 of 𝐴𝐡. Find the coordinates of 𝐢.

  • A 𝐢 = ( 0 , 8 )
  • B 𝐢 = ( βˆ’ 4 , βˆ’ 8 )
  • C 𝐢 = ( 8 , 0 )
  • D 𝐢 = ( 4 , βˆ’ 4 )
  • E 𝐢 = ( 7 , 0 )

Q23:

Two points 𝐴 and 𝐡 are at (βˆ’1,5) and (2,βˆ’4) respectively. Point 𝐢 lies on the line segment 𝐴𝐡 such that the lengths of 𝐴𝐢 and 𝐢𝐡 are in the ratio of 2∢1. Find the coordinates of 𝐢.

  • A 𝐢 = ( 1 , βˆ’ 1 )
  • B 𝐢 = ( 1 , 1 )
  • C 𝐢 = ( 1 , 0 )
  • D 𝐢 = ( 0 , βˆ’ 1 )
  • E 𝐢 = ( 0 , βˆ’ 3 )

Q24:

Two points 𝐴 and 𝐡 are at (βˆ’1,5) and (5,βˆ’1) respectively. Point 𝐢 lies on the line segment 𝐴𝐡 such that the lengths of 𝐴𝐢 and 𝐢𝐡 are in the ratio of 5∢1. Find the coordinates of 𝐢.

  • A 𝐢 = ( 2 , βˆ’ 2 )
  • B 𝐢 = ( 4 , 0 )
  • C 𝐢 = ( 4 , 1 )
  • D 𝐢 = ( βˆ’ 4 , 0 )
  • E 𝐢 = ( 0 , 4 )

Q25:

A quadrilateral has its vertices at the points 𝐴(βˆ’5,3), 𝐡(0,βˆ’2), 𝐢(βˆ’2,βˆ’6), and 𝐷(βˆ’8,βˆ’2). A point 𝐸 lies on 𝐴𝐢 such that the lengths of 𝐴𝐸 and 𝐢𝐸 are in the ratio of 1∢2, and a point 𝐹 lies on 𝐡𝐷 such that the lengths of 𝐡𝐹 and 𝐷𝐹 are in the ratio of 1∢3.

Find the coordinates of 𝐸.

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

Find the coordinates of 𝐹.

  • A ( βˆ’ 4 , βˆ’ 2 )
  • B ( βˆ’ 2 , βˆ’ 2 )
  • C ( βˆ’ 6 , βˆ’ 2 )
  • D ( βˆ’ 4 , 0 )
  • E ( βˆ’ 2 , 0 )

Find the slope of the line ⃖⃗𝐸𝐹.

Find the equation of the line ⃖⃗𝐸𝐹, giving your answer in the form 𝑦=π‘šπ‘₯+𝑐.

  • A 𝑦 = βˆ’ ( π‘₯ + 4 )
  • B 𝑦 = π‘₯ + 4
  • C 𝑦 = π‘₯ 4 + 1
  • D 𝑦 = 4 π‘₯ βˆ’ 4
  • E 𝑦 = π‘₯ + 1 4

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