Question

point n lies on $overline{lm}$ such that $ln:mn$ is $4:5$. graph n.

Explanation:

1. First, assume the coordinates of \(L(x_1,y_1)\) and \(M(x_2,y_2)\) from the graph:

• From the graph, \(L(6,19)\) and \(M(14,1)\).
• The section - formula for a point \(N(x,y)\) that divides the line - segment joining \(L(x_1,y_1)\) and \(M(x_2,y_2)\) in the ratio \(m:n\) is given by \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\), where \(m = 4\) and \(n = 5\).

1. Calculate the \(x\) - coordinate of \(N\):

• Substitute \(x_1 = 6\), \(x_2 = 14\), \(m = 4\), and \(n = 5\) into the \(x\) - coordinate formula \(x=\frac{mx_2+nx_1}{m + n}\).
• \(x=\frac{4\times14 + 5\times6}{4 + 5}=\frac{56+30}{9}=\frac{86}{9}\approx9.56\).

1. Calculate the \(y\) - coordinate of \(N\):

• Substitute \(y_1 = 19\), \(y_2 = 1\), \(m = 4\), and \(n = 5\) into the \(y\) - coordinate formula \(y=\frac{my_2+ny_1}{m + n}\).
• \(y=\frac{4\times1+5\times19}{4 + 5}=\frac{4 + 95}{9}=\frac{99}{9}=11\).

Step1: Identify coordinates of \(L\) and \(M\)

\(L(6,19)\), \(M(14,1)\)

Step2: Use section - formula for \(x\) - coordinate

\(x=\frac{4\times14 + 5\times6}{4 + 5}=\frac{56 + 30}{9}=\frac{86}{9}\approx9.56\)

Step3: Use section - formula for \(y\) - coordinate

\(y=\frac{4\times1+5\times19}{4 + 5}=\frac{4+95}{9}=11\)

Answer:

The coordinates of \(N\) are approximately \((9.56,11)\). Graph the point \(N\) on the given coordinate - plane at approximately \(x = 9.56\) and \(y = 11\).