Question

point b on a segment with endpoints a(2, -1) and c(4, 2) partitions the segment in a 1:3 ratio. find b.
(0.5, 0.75)
(-0.25, 2.5)
(0.75, 0.5)
(2.5, -0.25)

Explanation:

Step1: Recall section - formula

If a point \(B(x,y)\) divides the line - segment joining \(A(x_1,y_1)\) and \(C(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1 = 2,y_1=-1,x_2 = 4,y_2 = 2,m = 1,n = 3\).

Step2: Calculate the \(x\) - coordinate of \(B\)

\[

$$\begin{align*} x&=\frac{1\times4+3\times2}{1 + 3}\\ &=\frac{4 + 6}{4}\\ &=\frac{10}{4}=2.5 \end{align*}$$

\]

Step3: Calculate the \(y\) - coordinate of \(B\)

\[

$$\begin{align*} y&=\frac{1\times2+3\times(-1)}{1 + 3}\\ &=\frac{2-3}{4}\\ &=-\frac{1}{4}=-0.25 \end{align*}$$

\]

Answer:

D. \((2.5,-0.25)\)