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On a number line, the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] has endpoints [tex]\( Q \)[/tex] at [tex]\(-2\)[/tex] and [tex]\( S \)[/tex] at [tex]\( 6 \)[/tex]. Point [tex]\( R \)[/tex] partitions the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in a [tex]\( 3:2 \)[/tex] ratio. Rachel uses the section formula to find the location of point [tex]\( R \)[/tex] on the number line. Her work is shown below.

Let [tex]\( m = 3 \)[/tex], [tex]\( n = 2 \)[/tex], [tex]\( x_1 = -2 \)[/tex], and [tex]\( x_2 = 6 \)[/tex].

1. [tex]\( R = \frac{m x_2 + n x_1}{m + n} \)[/tex]
2. [tex]\( R = \frac{3(6) + 2(-2)}{3 + 2} \)[/tex]

What is the location of point [tex]\( R \)[/tex] on the number line?

A. [tex]\(\frac{14}{5}\)[/tex]

B. [tex]\(\frac{16}{5}\)[/tex]

C. [tex]\(\frac{18}{5}\)[/tex]

D. [tex]\(\frac{22}{5}\)[/tex]



Answer :

GinnyAnswer
To determine the location of point [tex]\( R \)[/tex] on the number line, we can use the section formula for a line segment divided in a given ratio. In this problem, point [tex]\( R \)[/tex] divides the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in a ratio of 3:2. Let's denote the given values:

- The coordinates of point [tex]\( Q \)[/tex] are [tex]\( x_1 = -2 \)[/tex].
- The coordinates of point [tex]\( S \)[/tex] are [tex]\( x_2 = 6 \)[/tex].
- The ratio [tex]\( m:n = 3:2 \)[/tex].

The section formula for a point dividing a line segment in a given ratio [tex]\( m:n \)[/tex] is given by:

[tex]\[ R = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \][/tex]

Substituting the given values into the formula:

[tex]\[ R = \frac{3 \cdot 6 + 2 \cdot (-2)}{3 + 2} \][/tex]

Next, we calculate the numerator and the denominator:

[tex]\[ R = \frac{18 + (-4)}{5} \][/tex]

Simplify the expression inside the fraction:

[tex]\[ R = \frac{18 - 4}{5} \][/tex]

[tex]\[ R = \frac{14}{5} \][/tex]

Thus, the location of point [tex]\( R \)[/tex] on the number line is:

[tex]\[ \boxed{\frac{14}{5}} \][/tex]

This confirms that point [tex]\( R \)[/tex] is located at [tex]\( \frac{14}{5} \)[/tex] on the number line.

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