Question

Directed Line Segment Worksheet Number Line Find the partitioning point for each problem. 1. Given the point A(-3) and B(6) find the coordinates of the point P on directed line segment AB that partitions AB in the ratio 2:1. 2. Given the points A(-3) and B(2) find the coordinates of the point P on directed line segment AB that partitions AB in the ratio 2 to 3. 3. Given the points A(-3) and B(4) find the coordinates of the point P on directed line segment AB that partitions AB in the ratio 5:2. 4. Given the point A(-9) and B(5) find the coordinates of the point P on directed line segment AB that partitions AB in the ratio 3:4. Coordinate Plane

Answer 1

Answer:

$1. P(3,0); 2. P(-1,-\frac{12}{5} ); 3. P(\frac{16}{7},\frac{27}{7} ); 4.P(-3,-6)$

Step-by-step explanation:

If coordinates of points are $A(x_1,y_1)$ and $B(x_2,y_2)$ and you need to divide the segment AB in the ratio m:n, where $\lambda=\frac{m}{n}$, then coordinates of point P which partitions segment AB in given ratio are:

$x=\frac{x_1+ \lambda x_2}{1+\lambda}, y=\frac{y_1+\lambda y_2}{1+\lambda}$

1. Here, $A(-3,-2), B(6,1), m:n=2:1,$ which means $\lambda=\frac{2}{1}=2$, so we get the following coordinates:

$x=\frac{-3+2*6}{1+2}=3 , y=\frac{-2+2*1}{1+2}=0$

We got that point P has coordinates P(3,0).  

2. Here, $A(-3,-4), B(2,0), m:n=2:3,$ which means $\lambda=\frac{2}{3},$  so, we get the following coordinates:

$x=\frac{-3+\frac{2}{3}*2 }{1+\frac{2}{3} }=\frac{\frac{-9+4}{3} }{\frac{5}{3} }=-1,\\y=\frac{-4+\frac{2}{3}*0 }{1+\frac{2}{3} }=\frac{-\frac{4}{1} }{\frac{5}{3} }=-\frac{12}{5}$

We got that point P has coordinates $P(-1,-\frac{12}{5} )$.

3.  Here, $A(-2,1), B(-4,5), m:n=5:2,$  which means $\lambda=\frac{5}{2}$, so we get the following coordinates:

$x=\frac{-2+\frac{5}{2}*4 }{1+\frac{5}{2} }=\frac{\frac{-4+20}{2} }{\frac{7}{2} }=\frac{16}{7},\\y=\frac{1+\frac{5}{2}*5 }{1+\frac{5}{2} }=\frac{\frac{2+25}{2} }{\frac{7}{2} }=\frac{27}{7}$

We got that point P has coordinates $P(\frac{16}{7} ,\frac{27}{7} )$.

4. Here, $A(-9,-9), B(5,-2), m:n=3:4,$ which means $\lambda=\frac{3}{4}$ so we get the following coordinates:

$x=\frac{-9+\frac{3}{4}*5 }{1+\frac{3}{4} }=\frac{-\frac{36+15}{4} }{\frac{7}{4} }=-3,\\y=\frac{-9+\frac{3}{4}(-2) }{1+\frac{3}{4} }=\frac{-\frac{36-6}{4} }{\frac{7}{4} }=-6$

We got that point P has coordinates P(-3,-6).