Question

no has endpoints n(4, 1) and o(13, 16). point p divides no into two parts with lengths in a ratio of 5:1.
what are the two possible locations of p?
(7.5, 12.5) (5.5, 3.5) (8.5, 8.5)
(11.5, 13.5) (5.8, 4) (12.5, 15.5)

Explanation:

Step1: Recall the section - formula

If a point $P(x,y)$ divides the line - segment joining $N(x_1,y_1)$ and $O(x_2,y_2)$ in the ratio $m:n$, then $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $x_1 = 4,y_1 = 1,x_2 = 13,y_2 = 16,m = 5,n = 1$.

Step2: Calculate the x - coordinate of $P$

$x=\frac{5\times13+1\times4}{5 + 1}=\frac{65 + 4}{6}=\frac{69}{6}=11.5$.

Step3: Calculate the y - coordinate of $P$

$y=\frac{5\times16+1\times1}{5 + 1}=\frac{80 + 1}{6}=\frac{81}{6}=13.5$.

Also, if we consider the ratio $n:m$ (i.e., $1:5$), then:

Step4: Calculate the alternative x - coordinate of $P$

$x=\frac{1\times13+5\times4}{1 + 5}=\frac{13+20}{6}=\frac{33}{6}=5.5$.

Step5: Calculate the alternative y - coordinate of $P$

$y=\frac{1\times16+5\times1}{1 + 5}=\frac{16 + 5}{6}=\frac{21}{6}=3.5$.

Answer:

$(5.5,3.5),(11.5,13.5)$