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Question
Math
Posted 2 months ago

Point AA is at (6,5)(-6,-5) and point CC is at (4,0)(4,0).
Find the coordinates of point BB on AC\overline{A C} such that the ratio of ABA B to BCB C is 2:32: 3.
B=(,)
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 BB on line segment AC\overline{AC} with a given ratio, we use the section formula
step 2
The section formula for a line divided in the ratio m:nm:n is B=(mx2+nx1m+n,my2+ny1m+n)B = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right), where A(x1,y1)A(x_1, y_1) and C(x2,y2)C(x_2, y_2)
step 3
Given A(6,5)A(-6, -5), C(4,0)C(4, 0), and the ratio AB:BC=2:3AB:BC = 2:3, we substitute m=2m=2, n=3n=3, x1=6x_1=-6, y1=5y_1=-5, x2=4x_2=4, and y2=0y_2=0 into the section formula
step 4
Calculate the x-coordinate of BB: Bx=24+3(6)2+3B_x = \frac{2 \cdot 4 + 3 \cdot (-6)}{2+3}
step 5
Calculate the y-coordinate of BB: By=20+3(5)2+3B_y = \frac{2 \cdot 0 + 3 \cdot (-5)}{2+3}
step 6
Simplify the coordinates: Bx=8185B_x = \frac{8 - 18}{5} and By=0155B_y = \frac{0 - 15}{5}
step 7
Final coordinates for BB: Bx=2B_x = -2 and By=3B_y = -3
[1] Answer
B=(2,3)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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