Question

in the coordinate plane, the directed line segment from k to n has endpoints at k (-6, -2) and n (8, 3). point l partitions the directed line segment from k to n in a ratio of 1 : 2. point m partitions the directed line segment from l to n in a ratio of 3 : 1. what are the coordinates of point m? round to the nearest tenth, if necessary. (-1.3, -0.3) (7, 2.5) (5.7, 2.2) (2.5, 1)

Explanation:

Step1: Find coordinates of L

The section formula for a point \((x,y)\) dividing a line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in ratio \(m:n\) is \(x=\frac{mx_2 + nx_1}{m + n}\), \(y=\frac{my_2 + ny_1}{m + n}\). For \(L\) dividing \(K(-6,-2)\) and \(N(8,3)\) in ratio \(1:2\) (\(m = 1\), \(n = 2\)):
\(x_L=\frac{1\times8+2\times(-6)}{1 + 2}=\frac{8-12}{3}=\frac{-4}{3}\approx - 1.3\)
\(y_L=\frac{1\times3+2\times(-2)}{1 + 2}=\frac{3-4}{3}=\frac{-1}{3}\approx - 0.3\)? Wait, no, wait: Wait, ratio \(1:2\) from \(K\) to \(N\), so \(m = 1\), \(n = 2\), so \(x=\frac{1\times8 + 2\times(-6)}{3}=\frac{8 - 12}{3}=\frac{-4}{3}\approx - 1.3\), \(y=\frac{1\times3+2\times(-2)}{3}=\frac{3 - 4}{3}=\frac{-1}{3}\approx - 0.3\)? But then \(L\) is \((-1.3, - 0.3)\)? Wait, no, maybe I mixed up \(m\) and \(n\). Wait, the ratio is \(1:2\) from \(K\) to \(N\), so the weights are \(m = 1\) (for \(N\)) and \(n = 2\) (for \(K\)). Wait, no, the formula is if the point is closer to \(K\) or \(N\). Wait, let's re - check. The section formula: if a point divides the line segment from \(A(x_1,y_1)\) to \(B(x_2,y_2)\) in the ratio \(k:1\) (i.e., \(m: n=k:1\)), then \(x=\frac{kx_2+x_1}{k + 1}\), \(y=\frac{ky_2+y_1}{k + 1}\). Here, the ratio is \(1:2\), so \(k=\frac{1}{2}\). So \(x_L=\frac{\frac{1}{2}\times8+(-6)}{\frac{1}{2}+1}=\frac{4 - 6}{\frac{3}{2}}=\frac{-2}{\frac{3}{2}}=\frac{-4}{3}\approx - 1.3\), \(y_L=\frac{\frac{1}{2}\times3+(-2)}{\frac{3}{2}}=\frac{\frac{3}{2}-2}{\frac{3}{2}}=\frac{-\frac{1}{2}}{\frac{3}{2}}=-\frac{1}{3}\approx - 0.3\). So \(L(-1.3, - 0.3)\)? Wait, no, that can't be right. Wait, maybe the ratio is \(1:2\) meaning \(KL:LN = 1:2\), so total parts \(3\). So \(L\) is \(\frac{1}{3}\) from \(K\) to \(N\). So \(x\) - coordinate: \(-6+\frac{1}{3}(8 - (-6))=-6+\frac{14}{3}=\frac{-18 + 14}{3}=\frac{-4}{3}\approx - 1.3\), \(y\) - coordinate: \(-2+\frac{1}{3}(3 - (-2))=-2+\frac{5}{3}=\frac{-6 + 5}{3}=-\frac{1}{3}\approx - 0.3\). So \(L(-1.3, - 0.3)\). Now, we need to find \(M\) which divides \(L(-1.3, - 0.3)\) to \(N(8,3)\) in ratio \(3:1\) (i.e., \(LM:MN = 3:1\)). So using the section formula again, with \(A = L(-1.3, - 0.3)\), \(B = N(8,3)\), ratio \(m:n = 3:1\). So \(x_M=\frac{3\times8+1\times(-1.3)}{3 + 1}=\frac{24-1.3}{4}=\frac{22.7}{4}=5.675\approx5.7\), \(y_M=\frac{3\times3+1\times(-0.3)}{3 + 1}=\frac{9 - 0.3}{4}=\frac{8.7}{4}=2.175\approx2.2\). So \(M(5.7,2.2)\).

Step2: Verify the calculation

We first found \(L\) by dividing \(K\) to \(N\) in \(1:2\) ratio. Then we used the coordinates of \(L\) and \(N\) to find \(M\) by dividing \(L\) to \(N\) in \(3:1\) ratio. The calculations for \(x\) and \(y\) coordinates of \(M\) using the section formula give us values that round to \(5.7\) and \(2.2\) respectively.

Answer:

C. \((5.7, 2.2)\)