Question

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 x - coordinate of point P>

The formula for the x - coordinate of a point \(P(x,y)\) that divides the line segment \(ST\) with \(S(x_1,y_1)\) and \(T(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2 + nx_1}{m + n}\).
For \(S(6,4)\), \(T(-4,-8)\) and ratio \(m = 1\), \(n=3\), we have \(x=\frac{1\times(-4)+3\times6}{1 + 3}=\frac{-4 + 18}{4}=\frac{14}{4}=\frac{7}{2}\).

Step2: <Calculate the y - coordinate of point P>

The formula for the y - coordinate is \(y=\frac{my_2+ny_1}{m + n}\).
Substituting \(y_1 = 4\), \(y_2=-8\), \(m = 1\), \(n = 3\) into the formula, we get \(y=\frac{1\times(-8)+3\times4}{1+3}=\frac{-8 + 12}{4}=\frac{4}{4}=1\).

Answer:

The coordinates of point \(P\) are \((\frac{7}{2},1)\)