Question

a line segment has endpoints ( a(1,4) ) and ( c(8,4) ). point ( b ) lies on ( overline{ac} ) and divides it such that the ratio ( ab:bc ) is equal to ( 4:3 ). what are the coordinates of ( b )?

Explanation:

Step1: Use the section formula

The section formula for a point \( B(x,y) \) that divides the line segment joining \( A(x_1,y_1) \) and \( C(x_2,y_2) \) in the ratio \( m:n \) is \( x=\frac{mx_2 + nx_1}{m + n} \), \( y=\frac{my_2+ny_1}{m + n} \). Here \( A(1,4) \), \( C(8,4) \), \( m = 4 \), \( n=3 \).

Step2: Calculate the \( x \) - coordinate

Substitute \( x_1 = 1 \), \( x_2=8 \), \( m = 4 \), \( n = 3 \) into \( x=\frac{mx_2+nx_1}{m + n} \).
\( x=\frac{4\times8+3\times1}{4 + 3}=\frac{32 + 3}{7}=\frac{35}{7}=5 \).

Step3: Calculate the \( y \) - coordinate

Substitute \( y_1 = 4 \), \( y_2 = 4 \), \( m = 4 \), \( n = 3 \) into \( y=\frac{my_2+ny_1}{m + n} \).
\( y=\frac{4\times4+3\times4}{4 + 3}=\frac{16+12}{7}=\frac{28}{7}=4 \).

Answer:

\((5,4)\)