What is the $x$-coordinate of the point that divides the directed line segment from J to K into a ratio of $2: 5$?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
Answer (2)
We have the formula x = ( m + n m ) ( x 2 − x 1 ) + x 1 .
Substitute m = 2 , n = 5 , x 1 = − 4 , and x 2 = 4 into the formula.
Calculate x = ( 2 + 5 2 ) ( 4 − ( − 4 )) + ( − 4 ) .
Simplify to find the x -coordinate: − 7 12 .
Explanation
Understanding the Formula We are given the formula to find the x -coordinate of a point that divides a directed line segment in a given ratio:
x = ( m + n m ) ( x 2 − x 1 ) + x 1
where:
m : n is the given ratio
x 1 is the x -coordinate of the starting point
x 2 is the x -coordinate of the ending point
Identifying Given Values We are given:
Ratio m : n = 2 : 5 , so m = 2 and n = 5
Starting point J has x 1 = − 4
Ending point K has x 2 = 4
Calculating the x-coordinate Substitute the given values into the formula:
x = ( 2 + 5 2 ) ( 4 − ( − 4 )) + ( − 4 )
x = ( 7 2 ) ( 4 + 4 ) − 4
x = ( 7 2 ) ( 8 ) − 4
x = 7 16 − 4
x = 7 16 − 7 28
x = 7 16 − 28
x = 7 − 12
x ≈ − 1.714
Final Answer The x -coordinate of the point that divides the directed line segment from J to K in the ratio 2 : 5 is:
x = − 7 12
Rounding to three decimal places, we get approximately -1.714.
Examples
In computer graphics, when drawing a line between two points, you might want to find a point that's a certain fraction of the way along that line. For instance, if you're creating a fading effect, you could use this formula to calculate the color at a point that's 2/7 of the way from one end of the line to the other, making the color gradually change.
The x -coordinate that divides the directed line segment from point J to point K in a ratio of 2 : 5 is − 7 12 , or approximately -1.714. This is calculated using a specific formula for dividing line segments in a given ratio.
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