Question

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. (-\frac{21}{5},\frac{23}{5}) b. (25,13) c. (-5,13) d. (25,22)

Explanation:

Step1: Use the section formula

If a point \(R(x,y)\) divides the line segment joining \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2 + nx_1}{m + n}\) 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:

$$4=\frac{2x_2+3\times(- 10)}{2 + 3}$$

$$4=\frac{2x_2-30}{5}$$

$$20=2x_2-30$$

$$2x_2=20 + 30$$

$$2x_2=50$$

$$x_2 = 25$$

Step2: For the \(y\) - coordinate

$$7=\frac{2y_2+3\times3}{2 + 3}$$

$$7=\frac{2y_2 + 9}{5}$$

$$35=2y_2+9$$

$$2y_2=35 - 9$$

$$2y_2=26$$

$$y_2 = 13$$

Answer:

B. \((25,13)\)