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.

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.

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.

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.

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.

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.

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.

This problem asks us to partition segment UV internally by a given ratio. With endpoints U(0,8) and V(12,-4), we need point P where UP:PV = 1:2. For internal division with ratio 1:2, P is located 1/3 of the way from U to V. Using the section formula: P = U + (1/(1+2))(V-U) = (0,8) + (1/3)(12,-12) = (0,8) + (4,-4) = (4,4). Alternatively, using weighted averages: x = (2(0) + 1(12))/(1+2) = 12/3 = 4, y = (2(8) + 1(-4))/3 = (16-4)/3 = 12/3 = 4. Therefore P = (4,4), which matches choice A. A common error would be to use ratio 2:1 instead, giving (8,0) as in choice C. The key is recognizing that P is closer to U than to V since the ratio is 1:2.

This problem asks for external division of a line segment. Given L(1,1) and M(5,9), we need point P where LP:PM = 1:3 with P beyond L opposite M. For external division with ratio 1:3, we use the formula P = (m·B - n·A)/(m-n) where m=1, n=3. Thus: x = (1(5) - 3(1))/(1-3) = (5-3)/(-2) = 2/(-2) = -1, y = (1(9) - 3(1))/(1-3) = (9-3)/(-2) = 6/(-2) = -3. Therefore P = (-1,-3), which matches choice A. This makes sense: P is beyond L in the direction opposite to M. A common error is using the internal division formula, which would give (4,7), not among the choices. The key insight is recognizing "external" division requires the modified formula with subtraction.

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.