The student's answer is actually incorrect. Let's analyze the student's work step-by-step and identify the correct approach.
1. Total Sections Calculation:
To partition the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in a ratio of [tex]\( 3:4 \)[/tex], we first need to determine the total number of sections:
[tex]\[
\text{total sections} = 3 + 4 = 7
\][/tex]
2. Fraction for the Partition:
We then use the fraction that corresponds to the given ratio. Since we want the point [tex]\( C \)[/tex] such that it partitions the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 3:4 \)[/tex]:
[tex]\[
\text{fraction} = \frac{3}{7}
\][/tex]
3. Calculate the Coordinate of Point [tex]\( C \)[/tex]:
The formula to find the coordinate of point [tex]\( C \)[/tex] is given by:
[tex]\[
C = \left( \frac{3}{7} \right)(B - A) + A
\][/tex]
Plugging in the given values [tex]\( A = -6 \)[/tex] and [tex]\( B = 2 \)[/tex]:
[tex]\[
C = \left( \frac{3}{7} \right)(2 - (-6)) + (-6)
\][/tex]
[tex]\[
C = \left( \frac{3}{7} \right)(8) + (-6)
\][/tex]
[tex]\[
C = \left( \frac{24}{7} \right) - 6
\][/tex]
[tex]\[
C = 3.4285714285714284 - 6
\][/tex]
[tex]\[
C = -2.5714285714285716
\][/tex]
4. Verification:
The correct position of point [tex]\( C \)[/tex] is approximately [tex]\(-2.57\)[/tex], not 0.
Thus, the student's error lies in the initial application of the ratio. They mistakenly used the fraction [tex]\( \frac{3}{4} \)[/tex] of the length instead of the correct fraction [tex]\( \frac{3}{7} \)[/tex].
So, the correct answer to your question is:
No, the student should have added [tex]\(3 + 4\)[/tex] to get the total number of sections, and used the fraction [tex]\( \frac{3}{7} \)[/tex] instead of [tex]\( \frac{3}{4} \)[/tex].
