Question

1. on a coordinate plane, a line segment has endpoints ( p(6,2) ) and ( q(3,8) ). point ( m ) lies on ( overline{pq} ) and divides the segment so that the ratio of ( pm:mq ) is ( 3:2 ). what are the coordinates of point ( m )?

a. ( (4.5,5.0) )
b. ( (5.6,4.2) )
c. ( (4.8,4.4) )
d. ( (4.2,5.6) )

Explanation:

Step1: Use the section formula

The section formula for a point \(M(x,y)\) 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, \(x_1 = 6,y_1 = 2,x_2 = 3,y_2 = 8,m = 3,n = 2\)

Step2: Calculate the \(x\) - coordinate of \(M\)

Substitute the values into the \(x\) - coordinate formula:
\(x=\frac{3\times3+2\times6}{3 + 2}=\frac{9 + 12}{5}=\frac{21}{5}=4.2\)

Step3: Calculate the \(y\) - coordinate of \(M\)

Substitute the values into the \(y\) - coordinate formula:
\(y=\frac{3\times8+2\times2}{3 + 2}=\frac{24+4}{5}=\frac{28}{5}=5.6\)

Answer:

D. \((4.2,5.6)\)