Question

Math

Posted 2 months ago

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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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Answer from Sia

Posted 2 months ago

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.

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