Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

Opening subject page...

Loading your content

Geometry

Geometry Help: Partitioning Line Segments Via Ratio

Review real example questions for Partitioning Line Segments Via Ratio in Geometry.

Question 1

In the coordinate plane shown, segment AB‾\overline{AB}AB has endpoints A(1,−3)A(1,-3)A(1,−3) and B(9,5)B(9,5)B(9,5). Point PPP divides the directed segment from AAA to BBB internally so that AP:PB=3:1AP:PB=3:1AP:PB=3:1. Which coordinates represent the partition point?

  1. (3,−1)(3,-1)(3,−1)
  2. (7,3)(7,3)(7,3)
  3. (5,1)(5,1)(5,1)
  4. (10,6)(10,6)(10,6)
Explanation: The skill is partitioning a line segment in a given ratio. The endpoints are A(1,-3) and B(9,5), with the ratio AP:PB = 3:1. This means point P is a weighted average where A has weight 1 and B has weight 3, since the weights are the opposite parts of the ratio. Applying the ratio, the coordinates are x = (11 + 39)/4 = 28/4 = 7 and y = (1*(-3) + 3*5)/4 = 12/4 = 3, so P is at (7,3). This result is justified because it positions P three-fourths of the way from A to B, consistent with the ratio 3:1. A common distractor misconception is using the midpoint, leading to (5,1), which ignores the unequal ratio. To transfer this strategy, think in terms of weights, not distances.

Question 2

On the coordinate plane, points A(−4,1)A(-4,1)A(−4,1) and B(2,7)B(2,7)B(2,7) are connected by segment AB‾\overline{AB}AB. Point PPP lies on the directed segment from AAA to BBB and divides AB‾\overline{AB}AB internally in the ratio AP:PB=1:2AP:PB=1:2AP:PB=1:2. Which point divides the segment in the given ratio?

  1. (−2,3)(-2,3)(−2,3)
  2. (0,5)(0,5)(0,5)
  3. (−6,−1)(-6,-1)(−6,−1)
  4. (−1,2)(-1,2)(−1,2)
Explanation: The skill is partitioning a line segment in a given ratio. The endpoints are A(-4,1) and B(2,7), with the ratio AP:PB = 1:2. This means point P is a weighted average where A has weight 2 and B has weight 1, since the weights are the opposite parts of the ratio. Applying the ratio, the coordinates are x = (2*(-4) + 12)/3 = -6/3 = -2 and y = (21 + 1*7)/3 = 9/3 = 3, so P is at (-2,3). This result is justified because it positions P one-third of the way from A to B, consistent with the ratio 1:2. A common distractor misconception is reversing the ratio to 2:1, leading to (0,5), which assumes the larger part is toward A instead of B. To transfer this strategy, think in terms of weights, not distances.

Question 3

In the coordinate plane, endpoints are A(−3,6)A(-3,6)A(−3,6) and B(9,0)B(9,0)B(9,0). Which coordinates represent the point PPP that divides AB‾\overline{AB}AB internally in the ratio AP:PB=5:1AP:PB=5:1AP:PB=5:1?

  1. (7,1)(7,1)(7,1)
  2. (6,2)(6,2)(6,2)
  3. (5,3)(5,3)(5,3)
  4. (8,0)(8,0)(8,0)
Explanation: This problem asks for point P that divides segment AB internally where A(-3,6) and B(9,0) with ratio AP:PB = 5:1. The endpoints are A(-3,6) and B(9,0), and P partitions the segment so that AP is 5 parts while PB is 1 part. Using the section formula with weighted averages, P = ((1·A + 5·B)/(5+1)), where we weight by opposite ratio parts. Calculating: P = ((1·(-3,6) + 5·(9,0))/6) = ((-3,6) + (45,0))/6 = (42,6)/6 = (7,1). This result is logical since P is very close to B, being 5/6 of the way from A to B. A typical mistake would be to use the ratio parts directly as weights, which would incorrectly place P near A. The key insight is that larger opposite weights pull the point toward that endpoint, so we weight B more heavily since P is closer to B.

Question 4

On the coordinate plane, segment AB‾\overline{AB}AB has endpoints A(0,2)A(0,2)A(0,2) and B(8,−6)B(8,-6)B(8,−6). Point PPP divides the directed segment from AAA to BBB internally in the ratio AP:PB=5:3AP:PB=5:3AP:PB=5:3. Which coordinates represent the partition point?

  1. (5,−3)(5,-3)(5,−3)
  2. (4,−2)(4,-2)(4,−2)
  3. (3,−1)(3,-1)(3,−1)
  4. (10,−8)(10,-8)(10,−8)
Explanation: The skill is partitioning a line segment in a given ratio. The endpoints are A(0,2) and B(8,-6), with the ratio AP:PB = 5:3. This means point P is a weighted average where A has weight 3 and B has weight 5, since the weights are the opposite parts of the ratio. Applying the ratio, the coordinates are x = (30 + 58)/8 = 40/8 = 5 and y = (32 + 5(-6))/8 = -24/8 = -3, so P is at (5,-3). This result is justified because it positions P five-eighths of the way from A to B, consistent with the ratio 5:3. A common distractor misconception is misapplying the weights, leading to (4,-2) by incorrectly averaging without proper ratio consideration. To transfer this strategy, think in terms of weights, not distances.

Question 5

Points N(−1,0)N(-1,0)N(−1,0) and O(7,8)O(7,8)O(7,8) are plotted on a coordinate plane and connected by segment NO‾\overline{NO}NO. Point VVV divides the directed segment from NNN to OOO internally in the ratio NV:VO=5:3NV:VO=5:3NV:VO=5:3. Which coordinates represent the partition point VVV?

  1. (4,5)(4,5)(4,5)
  2. (2,3)(2,3)(2,3)
  3. (3,4)(3,4)(3,4)
  4. (5,6)(5,6)(5,6)
Explanation: The skill here is partitioning a line segment in a given ratio using the section formula. The endpoints are N(-1,0) and O(7,8), with the ratio NV:VO = 5:3. This means point V is a weighted average of N and O, where the weight for N is 3 and for O is 5, because the weights are the ratios of the opposite segments. Applying the formula, the coordinates of V are ((5·7 + 3·(-1))/(5+3), (5·8 + 3·0)/(5+3)) = (4, 5). This result is justified because it places V such that the segment is divided into 5 parts from N to V and 3 parts from V to O, totaling 8 parts. A common distractor misconception is using equal weights, leading to the midpoint (3,4). To transfer this strategy, think in terms of weights assigned to each endpoint rather than direct distances.

Question 6

In the coordinate plane, segment AB‾\overline{AB}AB has endpoints A(2,5)A(2,5)A(2,5) and B(8,−1)B(8,-1)B(8,−1). Which point divides the directed segment AB→\overrightarrow{AB}AB internally in the ratio AP:PB=1:2AP:PB=1:2AP:PB=1:2?

  1. (4,3)(4,3)(4,3)
  2. (6,1)(6,1)(6,1)
  3. (5,2)(5,2)(5,2)
  4. (3,4)(3,4)(3,4)
Explanation: This problem requires finding point P that partitions segment AB where A(2,5) and B(8,-1) in the ratio AP:PB = 1:2. The endpoints are A(2,5) and B(8,-1), and P divides the segment so that AP is 1 part while PB is 2 parts of the total distance. Using the weighted average approach, we weight each endpoint by the opposite ratio part: P = ((2·A + 1·B)/(1+2)). Applying this formula: P = ((2·(2,5) + 1·(8,-1))/3) = ((4,10) + (8,-1))/3 = (12,9)/3 = (4,3). This result makes sense because P is closer to A than to B, being 1/3 of the way from A to B. A common mistake would be to weight A by 1 and B by 2, which would incorrectly place P closer to B. The key strategy is to remember that in weighted averages, larger weights pull the result closer to that point, so we use opposite ratio parts.

Question 7

Points A(0,0)A(0,0)A(0,0) and B(10,5)B(10,5)B(10,5) are plotted on the coordinate plane. Which point divides the directed segment AB→\overrightarrow{AB}AB internally in the ratio AP:PB=3:7AP:PB=3:7AP:PB=3:7?

  1. (3,2)(3,2)(3,2)
  2. (7,4)(7,4)(7,4)
  3. (5,2.5)(5,2.5)(5,2.5)
  4. (3,1.5)(3,1.5)(3,1.5)
Explanation: This problem requires finding point P that partitions segment AB where A(0,0) and B(10,5) in the ratio AP:PB = 3:7. The endpoints are A(0,0) and B(10,5), with P dividing the segment such that AP is 3 parts and PB is 7 parts of the total distance. Using the weighted average approach, P = ((7·A + 3·B)/(3+7)), weighting each endpoint by the opposite ratio part. Applying this: P = ((7·(0,0) + 3·(10,5))/10) = ((0,0) + (30,15))/10 = (30,15)/10 = (3,1.5). Since P is 3/10 of the way from A to B, it's much closer to A than to B, which matches our result. A common misconception would be to weight A by 3 and B by 7, incorrectly placing P closer to B. The transfer strategy is to visualize the ratio as weights in a balance, where opposite weights determine the equilibrium point.

Question 8

Point PPP is located at (−4,7)(-4, 7)(−4,7) and point QQQ is located at (8,−5)(8, -5)(8,−5). If point RRR divides segment PQ‾\overline{PQ}PQ​ in the ratio 2:12:12:1 from PPP to QQQ, and then point SSS divides segment PR‾\overline{PR}PR in the ratio 1:21:21:2 from PPP to RRR, what are the coordinates of point SSS?

  1. (0,3)(0, 3)(0,3)
  2. (2,1)(2, 1)(2,1)
  3. (−1,5)(-1, 5)(−1,5)
  4. (4,−1)(4, -1)(4,−1)
Explanation: First, find point R that divides PQ in ratio 2:1. Using the section formula: R = P + (2/3)(Q - P) = (-4, 7) + (2/3)((8, -5) - (-4, 7)) = (-4, 7) + (2/3)(12, -12) = (-4, 7) + (8, -8) = (4, -1). Next, find point S that divides PR in ratio 1:2. S = P + (1/3)(R - P) = (-4, 7) + (1/3)((4, -1) - (-4, 7)) = (-4, 7) + (1/3)(8, -8) = (-4, 7) + (8/3, -8/3) = (0, 3). Choice B incorrectly uses the wrong ratio for the second partition. Choice C results from switching the direction of one of the ratios. Choice D gives the coordinates of point R instead of point S.

Question 9

A directed line segment from point X(−6,4)X(-6, 4)X(−6,4) to point Y(9,−2)Y(9, -2)Y(9,−2) is partitioned by point ZZZ such that XZ‾:ZY‾=m:n\overline{XZ} : \overline{ZY} = m : nXZ:ZY=m:n where mmm and nnn are positive integers. If the xxx-coordinate of point ZZZ is 333, what is the value of mn\frac{m}{n}nm​?

  1. 23\frac{2}{3}32​
  2. 32\frac{3}{2}23​
  3. 35\frac{3}{5}53​
  4. 53\frac{5}{3}35​
Explanation: Using the section formula, if Z partitions XY in ratio m:n, then Z = (nX + mY)/(m + n). For the x-coordinate: 3 = (n(-6) + m(9))/(m + n) = (-6n + 9m)/(m + n). Cross-multiplying: 3(m + n) = -6n + 9m, so 3m + 3n = -6n + 9m, which gives 9n = 6m, or m/n = 9/6 = 3/2. Choice A reverses the ratio. Choice C would result from incorrectly setting up the equation. Choice D results from confusing which segment corresponds to which part of the ratio.

Question 10

On the coordinate plane, points L(−4,−1)L(-4,-1)L(−4,−1) and M(2,5)M(2,5)M(2,5) are connected by segment LM‾\overline{LM}LM. Which point divides the directed segment from LLL to MMM internally in the ratio LP:PM=1:2LP:PM=1:2LP:PM=1:2?

  1. (−1,2)(-1,2)(−1,2)
  2. (0,3)(0,3)(0,3)
  3. (−2,1)(-2,1)(−2,1)
  4. (−3,0)(-3,0)(−3,0)
Explanation: The skill involves partitioning a line segment in a given ratio using coordinates. Here, the endpoints are L(-4,-1) and M(2,5), and the ratio LP:PM is 1:2. The point P is a weighted average of the coordinates of L and M, with weights corresponding to the opposite segments: weight 2 for L and 1 for M. Thus, x = (2*(-4) + 12)/(1+2) = (-8 + 2)/3 = -6/3 = -2, and y = (2(-1) + 1*5)/3 = (-2 + 5)/3 = 3/3 = 1, so P is (-2,1). This result places P such that it divides the segment with LP being 1/3 and PM being 2/3 of the total length, satisfying the ratio. A common distractor misconception is swapping to 2:1, leading to (0,3), which is choice B. Transfer strategy: think in terms of weights, not distances.