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On a number line, the directed line segment from \( Q \) to \( S \) has endpoints \( Q \) at -8 and \( S \) at 12. The segment is divided in a \( 4:1 \) ratio.

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

A. \(\left(\frac{1}{1+4}\right)(12-(-8)) + (-8)\)
B. \(\left(\frac{4}{4+1}\right)(12-(-8)) + (-8)\)
C. \(\left(\frac{4}{4+1}\right)(-8-12) + 12\)
D. [tex]\(\left(\frac{4}{1+4}\right)(-8-12) + 12\)[/tex]



Answer :

GinnyAnswer
To solve the problem, we need to use the formula for finding a location that divides a line segment in a given ratio. The formula is:

[tex]\[ \left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 \][/tex]

Here, \(Q\) is at \(-8\) (which we'll denote as \(x_1\)), and \(S\) is at \(12\) (which we'll denote as \(x_2\)). The ratio provided is \(4:1\), which corresponds to \(m = 4\) and \(n = 1\).

We need to substitute these values into the formula:

[tex]\[ \left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 \][/tex]

Substituting \(m = 4\), \(n = 1\), \(x_1 = -8\), and \(x_2 = 12\), we get:

[tex]\[ \left(\frac{4}{4+1}\right)(12-(-8))+(-8) \][/tex]

First, calculate the denominator inside the fraction:

[tex]\[ 4 + 1 = 5 \][/tex]

So the formula becomes:

[tex]\[ \left(\frac{4}{5}\right)(12-(-8))+(-8) \][/tex]

Next, simplify inside the parentheses:

[tex]\[ 12 - (-8) = 12 + 8 = 20 \][/tex]

So now the formula is:

[tex]\[ \left(\frac{4}{5}\right)(20)+(-8) \][/tex]

Calculate the fraction times 20:

[tex]\[ \frac{4}{5} \times 20 = 16 \][/tex]

Add this result to \(-8\):

[tex]\[ 16 + (-8) = 8 \][/tex]

Therefore, the location that divides the directed line segment from \(Q\) to \(S\) in a \(4:1\) ratio is:

[tex]\[ 8 \][/tex]

The correct expression you need that matches the steps taken is:

[tex]\[ \left(\frac{4}{4+1}\right)(12-(-8))+(-8) \][/tex]

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