Question

find the coordinates of the point that partitions ab in the ratio 3:7. the point that partitions ab in the ratio 3:7 is (simplify your answer. type an ordered pair.)

Explanation:

Step1: Identify the coordinates of A and B

Let \(A(- 4,-6)\) and \(B(12,5)\).

Step2: Use the section - formula

The section formula for a point \(P(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 3\) and \(n = 7\), \(x_1=-4\), \(y_1=-6\), \(x_2 = 12\), \(y_2 = 5\).

Calculate the x - coordinate

\[

$$\begin{align*} x&=\frac{3\times12+7\times(-4)}{3 + 7}\\ &=\frac{36-28}{10}\\ &=\frac{8}{10}\\ &=\frac{4}{5} = 0.8 \end{align*}$$

\]

Calculate the y - coordinate

\[

$$\begin{align*} y&=\frac{3\times5+7\times(-6)}{3 + 7}\\ &=\frac{15-42}{10}\\ &=\frac{-27}{10}\\ &=-2.7 \end{align*}$$

\]

Answer:

\((0.8,-2.7)\)