Question

the coordinates of the endpoints of (overline{gh}) are (g(2, 16)) and (h(17, 7)). point (i) is on (overline{gh}) and divides it such that (gi:hi) is (2:1). what are the coordinates of (i)? write your answers as integers or decimals.

Explanation:

Step1: Recall the section formula

The section formula for a point \( I(x,y) \) dividing the line segment joining \( G(x_1,y_1) \) and \( H(x_2,y_2) \) in the ratio \( m:n \) is \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2+ny_1}{m + n} \). Here, \( m = 2 \), \( n=1 \), \( x_1 = 2 \), \( y_1=16 \), \( x_2 = 17 \), \( y_2 = 7 \).

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

Using the formula for \( x \) - coordinate:
\( x=\frac{2\times17+1\times2}{2 + 1}=\frac{34 + 2}{3}=\frac{36}{3}=12 \)

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

Using the formula for \( y \) - coordinate:
\( y=\frac{2\times7+1\times16}{2 + 1}=\frac{14+16}{3}=\frac{30}{3}=10 \)

Answer:

\( (12,10) \)