Question

the endpoints of the directed line segment ab are a(1,2) and b(10,5). find the coordinates of point p along segment ab so that the ratio of ap to pb is 1 to 2. coordinates: p ( , )

Explanation:

Step1: Use the section formula

If a point \(P(x,y)\) divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here \(m = 1\), \(n = 2\), \(x_1=1\), \(y_1 = 2\), \(x_2=10\), \(y_2=5\).

Step2: Calculate the \(x\) - coordinate

Substitute the values into the \(x\) - coordinate formula:
\(x=\frac{1\times10+2\times1}{1 + 2}=\frac{10 + 2}{3}=\frac{12}{3}=4\)

Step3: Calculate the \(y\) - coordinate

Substitute the values into the \(y\) - coordinate formula:
\(y=\frac{1\times5+2\times2}{1 + 2}=\frac{5+4}{3}=\frac{9}{3}=3\)

Answer:

\(P(4,3)\)