Question

3.5 reteach
a directed line segment ab is a segment that represents moving from point a to point b.
example partitioning a directed line segment
find the coordinates of point q along the directed line segment
cd so that the ratio of cq to qd is 1 to 3.
solution
in order to divide the segment in the ratio 1 to 3, think of dividing,
or partitioning, the segment into 1 + 3, or 4 congruent pieces.
point q is the point that is \\( \frac { 1 } { 4 } \\) of the way from point c to point d.
find the rise and run from point c to point d. leave the slope in terms of rise and run,
and do not simplify. let \\( ( x _ { 1 }, y _ { 1 } ) = ( 1,4 ) \\) and \\( ( x _ { 2 }, y _ { 2 } ) = ( 5,1 ) \\).
slope of \\( \overline { c d } : m = \frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } } = \frac { 1 - 4 } { 5 - 1 } = \frac { - 3 } { 4 } = \frac { \text { rise } } { \text { run } } \\)
to find the coordinates of point q, add \\( \frac { 1 } { 4 } \\) of the run to the x - coordinate of c,
and add \\( \frac { 1 } { 4 } \\) of the rise to the y - coordinate of c.
run: \\( \frac { 1 } { 4 } \\) of \\( 4 = \frac { 1 } { 4 } \cdot 4 = 1 \\) rise: \\( \frac { 1 } { 4 } \\) of \\( - 3 = \frac { 1 } { 4 } \cdot - 3 = - 0.75 \\)
so, the coordinates of q are \\( ( 1 + 1,4 + - 0.75 ) = ( 2,3.25 ) \\).
in exercises 1 and 2, find the coordinates of point p along the directed line segment
st so that sp to pt is the given ratio.

1. \\( s ( 6,4 ), t ( - 4, - 8 ) ; 1 \\) to 3 2. \\( s ( - 6,7 ), t ( 9,25 ) ; 2 \\) to 3

in exercises 3 and 4, find the coordinates of point q along the directed line segment lm so that lq to
qm is the given ratio.

1. \\( l ( - 1, - 2 ), m ( 3,6 ) ; 5 \\) to 3 4. \\( l ( 2,7 ), m ( - 1,1 ) ; 2 \\) to 1

Explanation:

Step1: Calculate the run and rise

Let \(S(x_1,y_1)=(-6,7)\) and \(T(x_2,y_2)=(9,25)\). The ratio \(SP:PT = 2:3\), so the total number of parts is \(2 + 3=5\).
The run (change in \(x\)) is \(x_2 - x_1=9-(-6)=15\). \(\frac{2}{5}\) of the run is \(\frac{2}{5}\times15 = 6\).
The rise (change in \(y\)) is \(y_2 - y_1=25 - 7 = 18\). \(\frac{2}{5}\) of the rise is \(\frac{2}{5}\times18=\frac{36}{5}=7.2\).

Step2: Find the coordinates of \(P\)

Add \(\frac{2}{5}\) of the run to the \(x\) - coordinate of \(S\) and \(\frac{2}{5}\) of the rise to the \(y\) - coordinate of \(S\).
The \(x\) - coordinate of \(P\) is \(x_1+\frac{2}{5}(x_2 - x_1)=-6 + 6=0\).
The \(y\) - coordinate of \(P\) is \(y_1+\frac{2}{5}(y_2 - y_1)=7+7.2 = 14.2\).

Answer:

The coordinates of point \(P\) are \((0,14.2)\)