Answered

On a number line, the directed line segment from $Q$ to $S$ has endpoints $Q$ at -14 and $S$ at 2. Point $R$ partitions the directed line segment from Q to S in a 3 : 5 ratio.



Which expression correctly uses the formula $\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$ to find the location of point R?


A$\left(\frac{3}{3+5}\right)(-14-2)+2$
B$\left(\frac{3}{3+5}\right)(-14-2)-2$
C$\left(\frac{3}{3+5}\right)(2-(-14))+(-14)$
D$\left(\frac{3}{3+5}\right)(2-14)+14$

Verified Answer :

  • Identify the variables from the problem: \(x_1 = -14\), \(x_2 = 2\), \(m = 3\), and \(n = 5\).
  • Substitute these into the formula \(\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1\) to get \(\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)\).
  • Compare the result with the given choices.
  • The correct choice is \(\boxed{c}\).

Explanation

  1. Analyzing the Problem and Formula

To find the location of point \(R\) on a directed line segment from \(Q\) to \(S\), we are given a standard partitioning formula:
\[\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1\]
In this problem, we are given:

  • The coordinate of endpoint \(Q\) is \(x_1 = -14\).
  • The coordinate of endpoint \(S\) is \(x_2 = 2\).
  • The partition ratio \(m:n\) is \(3:5\), which means \(m = 3\) and \(n = 5\).

Our goal is to correctly substitute these specific values into the formula to identify the correct expression for point \(R\).

  1. Substituting the Values

We now substitute the known values \(m=3\), \(n=5\), \(x_1=-14\), and \(x_2=2\) into the formula:
\[\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1\]
Substituting these values gives:
\[\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)\]
This expression represents the correct application of the partitioning formula. We can now compare this to the provided options.

  1. Identifying the Correct Choice

Comparing our derived expression \(\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)\) to the given choices:

A\(\left(\frac{3}{3+5}\right)(-14-2)+2\)
B\(\left(\frac{3}{3+5}\right)(-14-2)-2\)
C\(\left(\frac{3}{3+5}\right)(2-(-14)) + (-14)\)
D\(\left(\frac{3}{3+5}\right)(2-14)+14\)

The expression matches choice c. Therefore, choice c correctly uses the given formula to find the location of point \(R\).

Examples

Partitioning line segments is a fundamental concept in coordinate geometry with many practical applications. For instance, if you are designing a hiking trail on a map and want to place a rest station at a location that is two-thirds of the way between two markers, you would use this formula to calculate the exact coordinate of that station. This allows for precise planning in infrastructure development, mapping, and even in computer graphics where you need to interpolate positions between two points.

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