Point $A$ is at $(-6,-5)$ and point $C$ is at $(4,0)$.
Find the coordinates of point $B$ on $\overline{A C}$ such that the ratio of $A B$ to $B C$ is $2: 3$.
$$
B=(\square, \square)
$$

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Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the coordinates of point $B$ on line segment $\overline{AC}$ with a given ratio, we use the section formula. ‖ ‖ step 2 ⋮ The section formula for a line divided in the ratio $m:n$ is $B = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$, where $A(x_1, y_1)$ and $C(x_2, y_2)$. ‖ ‖ step 3 ⋮ Given $A(-6, -5)$, $C(4, 0)$, and the ratio $AB:BC = 2:3$, we substitute $m=2$, $n=3$, $x_1=-6$, $y_1=-5$, $x_2=4$, and $y_2=0$ into the section formula. ‖ ‖ step 4 ⋮ Calculate the x-coordinate of $B$: $B_x = \frac{2 \cdot 4 + 3 \cdot (-6)}{2+3}$. ‖ ‖ step 5 ⋮ Calculate the y-coordinate of $B$: $B_y = \frac{2 \cdot 0 + 3 \cdot (-5)}{2+3}$. ‖ ‖ step 6 ⋮ Simplify the coordinates: $B_x = \frac{8 - 18}{5}$ and $B_y = \frac{0 - 15}{5}$. ‖ ‖ step 7 ⋮ Final coordinates for $B$: $B_x = -2$ and $B_y = -3$. ‖ ∻[1] Answer∻ ⚹ $B=(-2, -3)$ ⚹ ∻Key Concept∻ ⚹Section Formula⚹ ∻Explanation∻ ⚹The section formula is used to find a point that divides a line segment into a given ratio.⚹◊What is the formula to find the coordinates of the midpoint of a line segment with endpoints $(-2, 4)$ and $(7, 3)$?⍭ Generate me a similar question◊

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