Question

qr has endpoints q(1, 3) and r(8, 10). point s divides qr into two parts with lengths in a ratio of 2:5.
what are the two possible locations of s?

Explanation:

Step1: Recall section - formula

If a point $S(x,y)$ divides the line - segment joining $Q(x_1,y_1)$ and $R(x_2,y_2)$ in the ratio $m:n$, then the coordinates of $S$ are given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $x_1 = 1,y_1 = 3,x_2 = 8,y_2 = 10,m = 2,n = 5$.

Step2: Calculate the x - coordinate of $S$

$x=\frac{2\times8+5\times1}{2 + 5}=\frac{16 + 5}{7}=\frac{21}{7}=3$.

Step3: Calculate the y - coordinate of $S$

$y=\frac{2\times10+5\times3}{2 + 5}=\frac{20+15}{7}=\frac{35}{7}=5$.
Also, if we consider the ratio $m = 5,n = 2$, then:

Step4: Calculate the new x - coordinate of $S$

$x=\frac{5\times8+2\times1}{5 + 2}=\frac{40 + 2}{7}=\frac{42}{7}=6$.

Step5: Calculate the new y - coordinate of $S$

$y=\frac{5\times10+2\times3}{5 + 2}=\frac{50+6}{7}=\frac{56}{7}=8$.

Answer:

$(3,5),(6,8)$