The coordinates of AB are A(-5,-1) and B(-2,-6). If AC: CB = 37, what are the coordinates of point C?
A. (-2.5,-1)
B. (-3.5,-2.8)
C. (-4.1,-2.5)
D. (-3.8,-3.2)
Answer (2)
The problem involves finding the coordinates of a point C that divides the line segment AB in a given ratio.
The section formula is used to calculate the coordinates of point C.
The x-coordinate of C is calculated as x c = 3 + 7 3 × ( − 2 ) + 7 × ( − 5 ) = − 4.1 .
The y-coordinate of C is calculated as y c = 3 + 7 3 × ( − 6 ) + 7 × ( − 1 ) = − 2.5 .
The coordinates of point C are ( − 4.1 , − 2.5 ) .
Explanation
Problem Analysis We are given the coordinates of points A and B, which are A(-5, -1) and B(-2, -6), respectively. We are also given that point C divides the line segment AB in the ratio AC:CB = 3:7. Our goal is to find the coordinates of point C.
Section Formula To find the coordinates of point C, we will use the section formula. The section formula states that if a point C divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of C are given by:
C = ( m + n m x 2 + n x 1 , m + n m y 2 + n y 1 )
Substitute Values In our case, we have:
A ( x 1 , y 1 ) = ( − 5 , − 1 ) B ( x 2 , y 2 ) = ( − 2 , − 6 ) m = 3 n = 7
Now, we substitute these values into the section formula to find the coordinates of point C.
Calculate x-coordinate Let's find the x-coordinate of C ( x c ):
x c = 3 + 7 3 × ( − 2 ) + 7 × ( − 5 ) = 10 − 6 − 35 = 10 − 41 = − 4.1
Calculate y-coordinate Now, let's find the y-coordinate of C ( y c ):
y c = 3 + 7 3 × ( − 6 ) + 7 × ( − 1 ) = 10 − 18 − 7 = 10 − 25 = − 2.5
Final Answer Therefore, the coordinates of point C are (-4.1, -2.5).
Examples
In architecture, determining the coordinates of a point that divides a line segment in a specific ratio is crucial for precise placement of structural elements. For instance, when designing a bridge, engineers need to accurately calculate the position of support beams relative to the endpoints of the bridge span to ensure stability and even weight distribution. The section formula helps achieve this precision, ensuring the structural integrity of the bridge.
The coordinates of point C that divides the line segment AB in the ratio AC:CB = 3:7 are (-4.1, -2.5). The correct answer is C. (-4.1, -2.5).
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