Question

the coordinates of the endpoints of \\(\overline{pq}\\) are \\(p(3, 5)\\) and \\(q(18, 15)\\). point \\(r\\) is on \\(\overline{pq}\\) and divides it such that \\(pr:qr\\) is \\(1:4\\). what are the coordinates of \\(r\\)? write your answers as integers or decimals. \\((\quad, \quad)\\)

Explanation:

Step1: Recall the section formula

When a point \( R(x,y) \) divides the line segment joining \( P(x_1,y_1) \) and \( Q(x_2,y_2) \) in the ratio \( m:n \), the coordinates of \( R \) are given by \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2+ny_1}{m + n} \). Here, \( m = 1 \), \( n = 4 \), \( x_1=3 \), \( y_1 = 5 \), \( x_2=18 \), \( y_2=15 \).

Step2: Calculate the x - coordinate of R

Using the formula for \( x \) - coordinate:
\( x=\frac{1\times18+4\times3}{1 + 4}=\frac{18 + 12}{5}=\frac{30}{5}=6 \)

Step3: Calculate the y - coordinate of R

Using the formula for \( y \) - coordinate:
\( y=\frac{1\times15+4\times5}{1 + 4}=\frac{15+20}{5}=\frac{35}{5}=7 \)

Answer:

\((6, 7)\)