Question

point x (\boxed{ }, \boxed{ }) is on segment yz so that $yx = \frac{3}{5}yz$. round your answers to the nearest tenth. y(-6,2) z(8,8)

Explanation:

Step1: Recall the section formula

The section formula for a point \( X(x,y) \) that divides the line segment joining \( Y(x_1,y_1) \) and \( Z(x_2,y_2) \) 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, since \( YX=\frac{3}{5}YZ \), the ratio \( m:n = 3:2 \) (because \( YX:XZ=3:2 \) as \( YX+XZ = YZ \) and \( YX=\frac{3}{5}YZ \) implies \( XZ=\frac{2}{5}YZ \)). Given \( Y(- 6,2)=(x_1,y_1) \) and \( Z(8,8)=(x_2,y_2) \), \( m = 3 \), \( n=2 \).

Step2: Calculate the x - coordinate of X

Using the formula for \( x \):
\( x=\frac{3\times8+2\times(-6)}{3 + 2}=\frac{24-12}{5}=\frac{12}{5}=2.4 \)

Step3: Calculate the y - coordinate of X

Using the formula for \( y \):
\( y=\frac{3\times8+2\times2}{3 + 2}=\frac{24 + 4}{5}=\frac{28}{5}=5.6 \)

Answer:

\( (2.4,5.6) \)