The student's methodology contains an error. Let's go through the process step-by-step to identify where the mistake lies.
1. Identifying the Points and the Total Number of Sections:
- Point [tex]\( A \)[/tex] is at -6 and Point [tex]\( B \)[/tex] is at 2.
- We need to partition the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 3:4 \)[/tex].
- The total number of sections created by this ratio is [tex]\( 3 + 4 = 7 \)[/tex].
2. Calculating the Specific Fraction of the Segment:
- The correct fraction that represents the position of point [tex]\( C \)[/tex] along the line segment [tex]\( AB \)[/tex] is [tex]\( \frac{3}{7} \)[/tex]. This fraction signifies that point [tex]\( C \)[/tex] is closer to [tex]\( A \)[/tex] than [tex]\( B \)[/tex], covering 3 parts out of the total 7 parts of the segment.
3. Finding the Distance Covered by the Fraction:
- Compute the distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]: since [tex]\( B - A = 2 - (-6) = 8 \)[/tex].
- The scaled distance for point [tex]\( C \)[/tex] using the fraction [tex]\( \frac{3}{7} \)[/tex] is calculated as [tex]\( \frac{3}{7} \times 8 = 3.4285714285714284 \)[/tex].
4. Determining the Position of Point [tex]\( C \)[/tex]:
- To find [tex]\( C \)[/tex], start at [tex]\( A \)[/tex] and move the scaled distance towards [tex]\( B \)[/tex]:
- Thus, point [tex]\( C \)[/tex] is located at [tex]\( A + \text{(fraction of the distance)} \)[/tex], which is [tex]\( -6 + \frac{3}{7} \times 8 \)[/tex].
- This simplifies to:
[tex]\[
-6 + 3.4285714285714284 = -2.5714285714285716
\][/tex]
Therefore, point [tex]\( C \)[/tex] is at approximately [tex]\(-2.57\)[/tex].
Analysis of the Student's Work:
- The student mistakenly used [tex]\( \frac{3}{4} \)[/tex] instead of [tex]\( \frac{3}{7} \)[/tex].
- Correcting the student's fractions and calculations reveals the accurate placement of [tex]\( C \)[/tex].
Thus, the student's approach is incorrect and the accurate calculations show that point [tex]\( C \)[/tex] is [tex]\( -2.57 \)[/tex].
