For the line segment whose endpoints are $A(0,0)$ and $B(4,3)$, find the $x$ value for the point located $\frac{2}{3}$ the distance from $A$ to $B$.
Answer (2)
Use the section formula to determine the coordinates of a point located a fraction of the distance along a line segment.
Apply the formula x = x A + 3 2 ( x B − x A ) to find the x-coordinate.
Substitute the given values x A = 0 and x B = 4 into the formula.
Calculate the x-coordinate: x = 3 8 ≈ 2.7 .
The x value is 2.7 .
Explanation
Problem Analysis We are given two points, A ( 0 , 0 ) and B ( 4 , 3 ) , and we want to find the x -coordinate of the point that is 3 2 of the distance from A to B . This problem involves finding a point along a line segment given a fraction of the distance.
Section Formula Let P ( x , y ) be the point that is 3 2 of the distance from A to B . We can use the section formula to find the coordinates of P . The section formula is given by:
P = A + 3 2 ( B − A )
X-coordinate Calculation To find the x -coordinate of point P , we use the formula:
x = x A + 3 2 ( x B − x A )
where x A and x B are the x -coordinates of points A and B , respectively.
Substitution and Simplification Substitute the given values x A = 0 and x B = 4 into the formula:
x = 0 + 3 2 ( 4 − 0 )
x = 3 2 × 4
x = 3 8
Final Calculation Now, we simplify the expression to find the x -coordinate:
x = 3 8 = 2.666...
Rounding to one decimal place, we get x ≈ 2.7 .
Final Answer The x value for the point located 3 2 the distance from A to B is 3 8 , which is approximately 2.7 .
Examples
Imagine you're designing a race track, and you need to place a checkpoint exactly 2/3 of the way along a straight section. If the starting point of the straight section is (0,0) and the end point is (4,3) on a coordinate grid, you can use this method to find the exact x-coordinate of where to place the checkpoint. This ensures accurate placement and fair timing for the race.
The x value for the point located 3 2 the distance from A ( 0 , 0 ) to B ( 4 , 3 ) is 3 8 , or approximately 2.67 . This result is calculated using the section formula for coordinates along a line segment. Thus, the final answer is 3 8 .
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