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Geometry Help: Partitioning Line Segments Via Ratio

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

Question 1 / 10

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

All questions

Question 1

$(3,-1)$
$(7,3)$ (correct answer)
$(5,1)$
$(10,6)$

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 $A(-4,1)$ and $B(2,7)$ are connected by segment $\overline{AB}$ . Point $P$ lies on the directed segment from $A$ to $B$ and divides $\overline{AB}$ internally in the ratio $AP:PB=1:2$ . Which point divides the segment in the given ratio?

$(-2,3)$ (correct answer)
$(0,5)$
$(-6,-1)$
$(-1,2)$

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

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

$(5,-3)$ (correct answer)
$(4,-2)$
$(3,-1)$
$(10,-8)$

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 4

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

$(4,5)$ (correct answer)
$(2,3)$
$(3,4)$
$(5,6)$

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 5

Point $P$ is located at $(-4, 7)$ and point $Q$ is located at $(8, -5)$ . If point $R$ divides segment $\overline{PQ}$ in the ratio $2:1$ from $P$ to $Q$ , and then point $S$ divides segment $\overline{PR}$ in the ratio $1:2$ from $P$ to $R$ , what are the coordinates of point $S$ ?

$(0, 3)$ (correct answer)
$(2, 1)$
$(-1, 5)$
$(4, -1)$

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 6

A directed line segment from point $X(-6, 4)$ to point $Y(9, -2)$ is partitioned by point $Z$ such that $\overline{XZ} : \overline{ZY} = m : n$ where $m$ and $n$ are positive integers. If the $x$ -coordinate of point $Z$ is $3$ , what is the value of $\frac{m}{n}$ ?

$\frac{2}{3}$
$\frac{3}{2}$ (correct answer)
$\frac{3}{5}$
$\frac{5}{3}$

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 7

Points $J(0,-4)$ and $K(10,6)$ are connected on a coordinate plane by segment $\overline{JK}$ . Point $T$ divides the directed segment from $J$ to $K$ internally in the ratio $JT:TK=4:1$ . Which point divides the segment in the given ratio?

$(8,4)$ (correct answer)
$(2,-2)$
$(5,1)$
$(4,0)$

Explanation: The skill here is partitioning a line segment in a given ratio using the section formula. The endpoints are J(0,-4) and K(10,6), with the ratio JT:TK = 4:1. This means point T is a weighted average of J and K, where the weight for J is 1 and for K is 4, because the weights are the ratios of the opposite segments. Applying the formula, the coordinates of T are ((4·10 + 1·0)/(4+1), (4·6 + 1·(-4))/(4+1)) = (8, 4). This result is justified because it places T such that the segment is divided into 4 parts from J to T and 1 part from T to K, totaling 5 parts. A common distractor misconception is swapping the ratio, leading to (2, -2) instead. To transfer this strategy, think in terms of weights assigned to each endpoint rather than direct distances.

Question 8

On the coordinate plane, segment $\overline{YZ}$ has endpoints $Y(-5,-2)$ and $Z(1,10)$ . Point $K$ lies on $\overline{YZ}$ and divides the directed segment from $Y$ to $Z$ internally in the ratio $YK:KZ=3:5$ . Which coordinates represent the partition point $K$ ?

$(-2,4)$
$(-\tfrac{11}{4},\tfrac{5}{2})$ (correct answer)
$(-1,6)$
$(-4,0)$

Explanation: The skill here is partitioning a line segment in a given ratio using the section formula. The endpoints are Y(-5,-2) and Z(1,10), with the ratio YK:KZ = 3:5. This means point K is a weighted average of Y and Z, where the weight for Y is 5 and for Z is 3, because the weights are the ratios of the opposite segments. Applying the formula, the coordinates of K are ((3·1 + 5·(-5))/(3+5), (3·10 + 5·(-2))/(3+5)) = (-11/4, 5/2). This result is justified because it places K such that the segment is divided into 3 parts from Y to K and 5 parts from K to Z, totaling 8 parts. A common distractor misconception is swapping the ratio, leading to (-1, 6) for 5:3 instead. To transfer this strategy, think in terms of weights assigned to each endpoint rather than direct distances.

Question 9

On the coordinate plane, segment $\overline{CD}$ has endpoints $C(1,5)$ and $D(9,1)$ . Point $Q$ divides the directed segment from $C$ to $D$ internally in the ratio $CQ:QD=1:3$ . Which point divides the segment in the given ratio?

$(3,4)$ (correct answer)
$(7,2)$
$(5,3)$
$(4,\tfrac{11}{3})$

Explanation: The skill here is partitioning a line segment in a given ratio using the section formula. The endpoints are C(1,5) and D(9,1), with the ratio CQ:QD = 1:3. This means point Q is a weighted average of C and D, where the weight for C is 3 and for D is 1, because the weights are the ratios of the opposite segments. Applying the formula, the coordinates of Q are ((1·9 + 3·1)/(1+3), (1·1 + 3·5)/(1+3)) = (3, 4). This result is justified because it places Q such that the segment is divided into 1 part from C to Q and 3 parts from Q to D, totaling 4 parts. A common distractor misconception is using the midpoint formula, leading to (5,3) instead. To transfer this strategy, think in terms of weights assigned to each endpoint rather than direct distances.

Question 10

Points $R(6,2)$ and $S(-3,-7)$ are connected by segment $\overline{RS}$ on a coordinate plane. Point $X$ divides the directed segment from $R$ to $S$ internally in the ratio $RX:XS=2:1$ . Which coordinates represent the partition point $X$ ?

$(0,-4)$ (correct answer)
$(3,-1)$
$(1,-3)$
$(\tfrac{3}{2},-\tfrac{5}{2})$

Explanation: The skill here is partitioning a line segment in a given ratio using the section formula. The endpoints are R(6,2) and S(-3,-7), with the ratio RX:XS = 2:1. This means point X is a weighted average of R and S, where the weight for R is 1 and for S is 2, because the weights are the ratios of the opposite segments. Applying the formula, the coordinates of X are ((2·(-3) + 1·6)/(2+1), (2·(-7) + 1·2)/(2+1)) = (0, -4). This result is justified because it places X such that the segment is divided into 2 parts from R to X and 1 part from X to S, totaling 3 parts. A common distractor misconception is using the midpoint, leading to (1.5, -2.5) or similar approximations. To transfer this strategy, think in terms of weights assigned to each endpoint rather than direct distances.