Question

find the coordinates of the point that is 3 times as close to a as it is to b. the point that is 3 times as close to a as it is to b is . (simplify your answer. type an ordered pair.)

Explanation:

Step1: Use section - formula

Let \(A(x_1,y_1)=(-4,-7)\) and \(B(x_2,y_2)=(12,5)\). If a point \(P(x,y)\) divides the line - segment joining \(A\) and \(B\) in the ratio \(m:n\), the section formula is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 3\) and \(n = 1\) since the point is 3 times as close to \(A\) as it is to \(B\).

Step2: Calculate the \(x\) - coordinate

\[

$$\begin{align*} x&=\frac{3\times12+1\times(-4)}{3 + 1}\\ &=\frac{36-4}{4}\\ &=\frac{32}{4}\\ & = 8 \end{align*}$$

\]

Step3: Calculate the \(y\) - coordinate

\[

$$\begin{align*} y&=\frac{3\times5+1\times(-7)}{3 + 1}\\ &=\frac{15-7}{4}\\ &=\frac{8}{4}\\ &=2 \end{align*}$$

\]

Answer:

\((8,2)\)