Let’s analyze the student’s work and correct it step by step.
The task is to find point [tex]\( C \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] such that the segment is partitioned in the ratio [tex]\( 3:4 \)[/tex]. Given points are [tex]\( A = -6 \)[/tex] and [tex]\( B = 2 \)[/tex].
First, let's determine the correct way to approach this:
1. Determine the Total Number of Sections:
The segment is divided in the ratio [tex]\( 3:4 \)[/tex]:
[tex]\[
\text{Total sections} = 3 + 4 = 7
\][/tex]
2. Calculate the Length of Each Section:
The distance between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[
B - A = 2 - (-6) = 8
\][/tex]
3. Find the Fraction of the Distance for Point [tex]\( C \)[/tex]:
Point [tex]\( C \)[/tex] needs to be located at a position that divides the segment in the ratio [tex]\( 3:4 \)[/tex]. Therefore, we use the fraction [tex]\( \frac{3}{7} \)[/tex]:
[tex]\[
C = A + \left( \frac{3}{7} \right) \times (B - A)
\][/tex]
4. Substitute the Values:
Substitute [tex]\( A = -6 \)[/tex], [tex]\( B = 2 \)[/tex], and the fraction [tex]\( \frac{3}{7} \)[/tex]:
[tex]\[
C = -6 + \left( \frac{3}{7} \right) \times (2 - (-6))
\][/tex]
[tex]\[
C = -6 + \left( \frac{3}{7} \right) \times 8
\][/tex]
[tex]\[
C = -6 + \left( \frac{3}{7} \right) \times 8
\][/tex]
[tex]\[
C = -6 + \frac{24}{7}
\][/tex]
[tex]\[
C = -6 + 3.4285714285714286
\][/tex]
[tex]\[
C \approx -2.5714285714285716
\][/tex]
According to the correct method, the coordinate of point [tex]\( C \)[/tex] should be approximately [tex]\(-2.5714285714285716\)[/tex].
Now let’s analyze the student’s work:
1. [tex]\[
c = \left( \frac{3}{4} \right)(2 -(-6)) + (-6)
\][/tex]
The student incorrectly used the fraction [tex]\( \frac{3}{4} \)[/tex] instead of [tex]\( \frac{3}{7} \)[/tex].
2. [tex]\[
c = \left( \frac{3}{4} \right)(8) - 6
\][/tex]
Correct intermediate step, given their wrong fraction.
3. [tex]\[
c = 6 - 6
\][/tex]
Calculated the term in parenthesis but still based on a wrong fraction.
4. [tex]\[
c = 0
\][/tex]
The final answer is incorrect.
Thus, the student's answer is not correct. They 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]. The correct coordinate for point [tex]\( C \)[/tex] is approximately [tex]\(-2.5714285714285716\)[/tex].
