To solve this problem accurately, you need to partition the line segment from point [tex]\( A \)[/tex] to point [tex]\( B \)[/tex] in the ratio [tex]\( 3:4 \)[/tex]. Here’s an appropriate step-by-step solution:
1. Point [tex]\( A \)[/tex] is at -6 and point [tex]\( B \)[/tex] is at 2.
2. The ratio given is [tex]\( 3:4 \)[/tex]. In order to find point [tex]\( C \)[/tex] that divides the segment [tex]\( AB \)[/tex] in this ratio, we need to use the fraction [tex]\( \frac{3}{3+4} = \frac{3}{7} \)[/tex].
Using the formula for the section of a line segment in a given ratio:
[tex]\[
C = A + \left(\frac{3}{3+4}\right) \times (B - A)
\][/tex]
Here’s the calculation step-by-step:
1. Calculate the total distance between [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ B - A = 2 - (-6) = 2 + 6 = 8 \][/tex]
2. Multiply this distance by the ratio [tex]\( \frac{3}{7} \)[/tex]:
[tex]\[ \frac{3}{7} \times 8 = \frac{24}{7} \approx 3.4285714285714284 \][/tex]
3. Add this result to point [tex]\( A \)[/tex]:
[tex]\[ A + \frac{24}{7} = -6 + \frac{24}{7} = - \frac{42}{7} + \frac{24}{7} = - \frac{42 - 24}{7} = - \frac{18}{7} = -2.5714285714285716\][/tex]
Therefore, the correct position of point [tex]\( C \)[/tex] is approximately [tex]\(-2.5714285714285716\)[/tex].
The student's work contained the following mistakes:
- The student used the incorrect formula; he/she used [tex]\( \frac{3}{4} \)[/tex] instead of the correct fraction [tex]\( \frac{3}{7} \)[/tex] for the partitioning ratio.
- The student subtracted 6, which is an incorrect approach because the correct approach is to simply add the resulting value to [tex]\( A \)[/tex].
So, the student's answer is not correct, and the proper method involves dividing by the sum of the parts of the ratio and then adding this to point [tex]\( A \)[/tex].
