Question

find the point that lies at two - fifths the length of the directed line segment from the point (1, 1) to the point (11, 6).
(5,3)
(3,2)
(9,5)
(6,3.5)

Explanation:

Step1: Recall the section formula

The section formula for a point \( P(x,y) \) that divides the line segment joining \( A(x_1,y_1) \) and \( B(x_2,y_2) \) in the ratio \( m:n \) is \( x=\frac{mx_2 + nx_1}{m + n} \), \( y=\frac{my_2+ny_1}{m + n} \). Here, the ratio \( m:n = 2:3 \) (since the point is at two - fifths the length from \( A(1,1) \) to \( B(11,6) \), so the remaining part is \( 1-\frac{2}{5}=\frac{3}{5} \), so \( m = 2 \), \( n = 3 \)), \( x_1=1,y_1 = 1,x_2=11,y_2=6 \).

Step2: Calculate the x - coordinate

Substitute the values into the formula for \( x \):
\( x=\frac{2\times11+3\times1}{2 + 3}=\frac{22 + 3}{5}=\frac{25}{5}=5 \)

Step3: Calculate the y - coordinate

Substitute the values into the formula for \( y \):
\( y=\frac{2\times6+3\times1}{2+3}=\frac{12 + 3}{5}=\frac{15}{5}=3 \)

Answer:

(5,3)