Question

the coordinates of the endpoints of \\(\overline{hi}\\) are \\(h(1, 4)\\) and \\(i(13, 16)\\). point \\(j\\) is on \\(\overline{hi}\\) and divides it such that \\(hj:ij\\) is \\(1:3\\). what are the coordinates of \\(j\\)? write your answers as integers or decimals. (\\(\square\\), \\(\square\\))

Explanation:

Step1: Recall the section formula

The section formula for a point \( J(x,y) \) that divides the line segment joining \( H(x_1,y_1) \) and \( I(x_2,y_2) \) in the ratio \( m:n \) is given by:
\[
x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}
\]
Here, \( H(1,4) \), \( I(13,16) \), and the ratio \( HJ:IJ = 1:3 \), so \( m = 1 \), \( n = 3 \), \( x_1=1 \), \( y_1 = 4 \), \( x_2=13 \), \( y_2=16 \).

Step2: Calculate the x - coordinate of J

Substitute the values into the formula for \( x \):
\[
x=\frac{1\times13+3\times1}{1 + 3}=\frac{13 + 3}{4}=\frac{16}{4}=4
\]

Step3: Calculate the y - coordinate of J

Substitute the values into the formula for \( y \):
\[
y=\frac{1\times16+3\times4}{1 + 3}=\frac{16+12}{4}=\frac{28}{4}=7
\]

Answer:

\((4, 7)\)