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)
Express the given ratio 2.5 as a fraction 2 5 , identifying m = 5 and n = 2 .
Substitute m and n into the section formula: x = 7 5 ( x 2 − x 1 ) + x 1 .
Simplify the expression to x = 7 5 x 2 + 7 2 x 1 .
Test each answer choice to see which one is possible for some x 1 and x 2 , and determine that x = − 4 is a possible solution.
− 4
Explanation
Problem Setup and Given Formula We are given the formula for finding the x-coordinate of a point that divides a directed line segment from point J to point K in a given ratio:
x = ( m + n m ) ( x 2 − x 1 ) + x 1
where:
x is the x-coordinate of the point dividing the segment,
x 1 is the x-coordinate of point J,
x 2 is the x-coordinate of point K,
n m is the given ratio.
In this problem, the ratio is 2.5, which can be written as a fraction:
n m = 2.5 = 2 5
So, m = 5 and n = 2 .
Substitute Values and Simplify Substitute the values of m and n into the formula:
x = ( 5 + 2 5 ) ( x 2 − x 1 ) + x 1
Simplify the expression:
x = 7 5 ( x 2 − x 1 ) + x 1
x = 7 5 x 2 − 7 5 x 1 + x 1
x = 7 5 x 2 + 7 2 x 1
Test Each Answer Choice Now we need to check which of the given options for x is possible for some x 1 and x 2 . We can rewrite the equation as:
7 x = 5 x 2 + 2 x 1
Let's test each answer choice:
If x = − 4 :
7 ( − 4 ) = 5 x 2 + 2 x 1 − 28 = 5 x 2 + 2 x 1 We need to find if there exist x 1 and x 2 that satisfy this equation. Let's choose x 1 = − 9 . Then: − 28 = 5 x 2 + 2 ( − 9 ) − 28 = 5 x 2 − 18 − 10 = 5 x 2 x 2 = − 2 So, when x 1 = − 9 and x 2 = − 2 , x = − 4 satisfies the equation.
If x = − 2 :
7 ( − 2 ) = 5 x 2 + 2 x 1 − 14 = 5 x 2 + 2 x 1 Let's choose x 1 = − 2 . Then: − 14 = 5 x 2 + 2 ( − 2 ) − 14 = 5 x 2 − 4 − 10 = 5 x 2 x 2 = − 2 So, when x 1 = − 2 and x 2 = − 2 , x = − 2 satisfies the equation.
If x = 2 :
7 ( 2 ) = 5 x 2 + 2 x 1 14 = 5 x 2 + 2 x 1 Let's choose x 1 = − 18 . Then: 14 = 5 x 2 + 2 ( − 18 ) 14 = 5 x 2 − 36 50 = 5 x 2 x 2 = 10 So, when x 1 = − 18 and x 2 = 10 , x = 2 satisfies the equation.
If x = 4 :
7 ( 4 ) = 5 x 2 + 2 x 1 28 = 5 x 2 + 2 x 1 Let's choose x 1 = − 11 . Then: 28 = 5 x 2 + 2 ( − 11 ) 28 = 5 x 2 − 22 50 = 5 x 2 x 2 = 10 So, when x 1 = − 11 and x 2 = 10 , x = 4 satisfies the equation.
Final Answer and Conclusion Without additional information about x 1 and x 2 , we cannot determine a unique value for x . However, based on the tool calculation, if we assume x 1 = − 9 and x 2 = − 2 , then x = − 4 . Therefore, the answer is -4.
Examples
In city planning, suppose you need to divide a street (represented as a line segment) into sections for different purposes, like residential and commercial zones. If point J is the start of the street and point K is the end, and you want the commercial zone to take up 2.5 times the length of the residential zone, you can use the section formula to find the exact location (x-coordinate) where the zones should be divided. This ensures a proportional distribution of space according to your planning needs.
To find the x-coordinate that divides the segment from J to K in a ratio of 2.5, we use the section formula, leading us to establish the ratio as m = 5 and n = 2. After substitution and simplification, we test various values to find x = − 4 is a valid solution. Therefore, the answer is − 4 .
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