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 traveling 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), (βˆ’2,2)
  • C(1,βˆ’1), (4,βˆ’4)
  • D(2,βˆ’2), (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)
  • Bο€Ό23,2, ο€Ό236,βˆ’12
  • C(1,3), (1,3)
  • Dο€Ό43,4, ο€Ό23,2

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’11)
  • C(βˆ’13,16)
  • 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(14,15)
  • 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(51,21), (βˆ’75,3)
  • Bο€Όβˆ’517,βˆ’3, (βˆ’75,3)
  • C(βˆ’51,βˆ’21), (βˆ’75,3)
  • Dο€Όβˆ’5111,βˆ’2111, ο€Όβˆ’757,37

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’2,10)
  • 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(11,βˆ’2)
  • B(14,7)
  • C(28,14)
  • D(22,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(12,βˆ’9)
  • B(4,βˆ’3)
  • C(6,βˆ’5)
  • D(18,βˆ’15)

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.

  • A3∢8Β internally, (0,12)
  • B3∢2Β internally, (βˆ’6,0)
  • C3∢2Β externally, (βˆ’6,0)
  • D3∢8Β 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⟩
  • DβŸ¨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.

  • A2∢7 externally, (βˆ’10,0)
  • B3∢2 externally, (βˆ’5,0)
  • C2∢7 internally, (βˆ’10,0)
  • D3∢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.

  • A2∢5Β  externally, 𝑦=βˆ’1
  • B5∢2Β  externally, 𝑦=1
  • C2∢5Β  internally, 𝑦=βˆ’1
  • D5∢2Β  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⟨22,βˆ’17⟩
  • B⟨17,22⟩
  • C⟨18,βˆ’20⟩
  • DβŸ¨βˆ’17,22⟩

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(23,βˆ’13)
  • B(8,βˆ’19)
  • C(βˆ’19,8)
  • D(19,8)

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.

  • A𝐢=(1,2)
  • B𝐢=(2,1)
  • C𝐢=(βˆ’1,βˆ’1)
  • D𝐢=(1,1)
  • E𝐢=(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.

  • A𝐢=(0,8)
  • B𝐢=(βˆ’4,βˆ’8)
  • C𝐢=(8,0)
  • D𝐢=(4,βˆ’4)
  • E𝐢=(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.

  • A𝐢=(1,βˆ’1)
  • B𝐢=(1,1)
  • C𝐢=(1,0)
  • D𝐢=(0,βˆ’1)
  • E𝐢=(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.

  • A𝐢=(2,βˆ’2)
  • B𝐢=(4,0)
  • C𝐢=(4,1)
  • D𝐢=(βˆ’4,0)
  • E𝐢=(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.

  • A𝑦=βˆ’(π‘₯+4)
  • B𝑦=π‘₯+4
  • C𝑦=π‘₯4+1
  • D𝑦=4π‘₯βˆ’4
  • E𝑦=π‘₯+14

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