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Line segment [tex]\(\overline{AB}\)[/tex] has endpoints [tex]\(A(-4, -10)\)[/tex] and [tex]\(B(-11, -7)\)[/tex].

Using the formula [tex]\(x = \left(\frac{a}{a+b}\right)(x_2 - x_1) + x_1\)[/tex], the [tex]\(x\)[/tex]-coordinate of the point that divides [tex]\(\overline{AB}\)[/tex] in a [tex]\(3:4\)[/tex] ratio is calculated as follows:

[tex]\[ x = \left(\frac{3}{3+4}\right)(-11 - (-4)) + (-4) \][/tex]

What is the [tex]\(x\)[/tex]-coordinate of the point that divides [tex]\(\overline{AB}\)[/tex] into a [tex]\(3:4\)[/tex] ratio?

A. [tex]\(-7\)[/tex]
B. [tex]\(-5\)[/tex]
C. [tex]\(-3\)[/tex]
D. [tex]\(-1\)[/tex]



Answer :

GinnyAnswer
To find the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment [tex]\(\overline{AB}\)[/tex] in a given ratio, we used the formula:

[tex]\[ x = \left(\frac{a}{a+b}\right)(x_2 - x_1) + x_1 \][/tex]

Here are the specific values we are working with:
- The coordinates of [tex]\( A \)[/tex] are [tex]\( (-4, -10) \)[/tex].
- The coordinates of [tex]\( B \)[/tex] are [tex]\( (-11, -7) \)[/tex].
- The ratio [tex]\( a:b \)[/tex] is given as [tex]\( 3:4 \)[/tex].

Substituting these values into our formula, we get:

[tex]\[ x = \left( \frac{3}{3+4} \right) (-11 - (-4)) + (-4) \][/tex]

Let's simplify this step-by-step:

1. Compute the denominator of the fraction: [tex]\( 3 + 4 = 7 \)[/tex].
2. Compute the difference in the [tex]\( x \)[/tex]-coordinates: [tex]\( -11 - (-4) = -11 + 4 = -7 \)[/tex].
3. Multiply the ratio fraction by this difference: [tex]\( \frac{3}{7} \times (-7) = -3 \)[/tex].
4. Add this result to the [tex]\( x \)[/tex]-coordinate of point [tex]\( A \)[/tex]: [tex]\( -3 + (-4) = -7 \)[/tex].

Thus, the [tex]\( x \)[/tex]-coordinate of the point that divides [tex]\(\overline{AB}\)[/tex] into a [tex]\( 3:4 \)[/tex] ratio is:

[tex]\[ \boxed{-7} \][/tex]

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