What are the $x$- and $y$-coordinates of point $C$, which partitions the directed line segment from A to B into the ratio 5:8? Round to the nearest tenth, if necessary.
[tex]x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1[/tex]
[tex]y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1[/tex]
A. (-2, 2, -6.3)
B. (-2.4, -6.4)
C. (2.7, -0.7)
D. (1.2, -4.7)
Answer (2)
Identify the given values: m = 5 , n = 8 , x 1 = − 2.2 , x 2 = 2.7 , y 1 = − 6.3 , y 2 = − 0.7 .
Calculate the x-coordinate of point C using the formula: x = ( 5 + 8 5 ) ( 2.7 − ( − 2.2 )) + ( − 2.2 ) ≈ − 0.3154 .
Calculate the y-coordinate of point C using the formula: y = ( 5 + 8 5 ) ( − 0.7 − ( − 6.3 )) + ( − 6.3 ) ≈ − 4.1462 .
Round the calculated coordinates to the nearest tenth and state the coordinates of point C: ( − 0.3 , − 4.1 ) .
Explanation
Problem Analysis and Given Data We are given the coordinates of points A and B, and we want to find the coordinates of point C that partitions the directed line segment from A to B in the ratio 5:8. The coordinates of point A are (-2.2, -6.3), and the coordinates of point B are (2.7, -0.7). The section ratio is given as 5:8. We will use the section formula to find the coordinates of point C.
Calculate the x-coordinate The section formula for the x-coordinate of point C is given by: x = ( m + n m ) ( x 2 − x 1 ) + x 1 where m = 5 , n = 8 , x 1 = − 2.2 , and x 2 = 2.7 . Substituting these values into the formula, we get: x = ( 5 + 8 5 ) ( 2.7 − ( − 2.2 )) + ( − 2.2 ) x = ( 13 5 ) ( 2.7 + 2.2 ) − 2.2 x = ( 13 5 ) ( 4.9 ) − 2.2 x = 13 24.5 − 2.2 x ≈ 1.8846 − 2.2 x ≈ − 0.3154
Calculate the y-coordinate The section formula for the y-coordinate of point C is given by: y = ( m + n m ) ( y 2 − y 1 ) + y 1 where m = 5 , n = 8 , y 1 = − 6.3 , and y 2 = − 0.7 . Substituting these values into the formula, we get: y = ( 5 + 8 5 ) ( − 0.7 − ( − 6.3 )) + ( − 6.3 ) y = ( 13 5 ) ( − 0.7 + 6.3 ) − 6.3 y = ( 13 5 ) ( 5.6 ) − 6.3 y = 13 28 − 6.3 y ≈ 2.1538 − 6.3 y ≈ − 4.1462
Round to the nearest tenth We are asked to round the coordinates to the nearest tenth. Therefore, x ≈ − 0.3 y ≈ − 4.1 So, the coordinates of point C are approximately (-0.3, -4.1).
Final Answer The coordinates of point C, which partitions the directed line segment from A to B in the ratio 5:8, are approximately (-0.3, -4.1).
Examples
In architecture, when designing a building facade, you might want to divide a vertical line segment representing the height of the facade into specific ratios to place windows or decorative elements. For example, if you want to place a decorative band that divides the facade in a 5:8 ratio, you can use the section formula to find the exact coordinates (height) where the band should be placed. This ensures precise and aesthetically pleasing placement of architectural elements.
The coordinates of point C that partition the directed line segment from A to B in the ratio 5:8 are approximately (-0.3, -4.1).
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