Question

5
consider directed line segment pq. point p is located at (-10,3). point r, which is on segment pq and divides segment pq into a ratio of pr:rq = 2:3,
is located at (4,7).
what are the coordinates of point q?
a. (25, 13)
b. (25, 22)
c. (-5, 13)
d. (-\frac{22}{5}, \frac{23}{5})

Explanation:

Step1: Use the section formula

The section formula for a point \(R(x,y)\) that divides the line segment joining \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) in the ratio \(m:n\) is given by \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 2\), \(n = 3\), \(x_1=-10\), \(y_1 = 3\), \(x = 4\), \(y = 7\).
For the \(x\)-coordinate:
\(4=\frac{2x_2+3\times(-10)}{2 + 3}\)
\(4=\frac{2x_2-30}{5}\)
Multiply both sides by \(5\): \(20=2x_2-30\)
Add \(30\) to both sides: \(2x_2=20 + 30=50\)
Divide by \(2\): \(x_2 = 25\)

Step2: For the \(y\)-coordinate

\(7=\frac{2y_2+3\times3}{2 + 3}\)
\(7=\frac{2y_2 + 9}{5}\)
Multiply both sides by \(5\): \(35=2y_2+9\)
Subtract \(9\) from both sides: \(2y_2=35 - 9=26\)
Divide by \(2\): \(y_2 = 13\)

Answer:

A. \((25,13)\)