A student wants to find point $C$ on the directed line segment from $A$ to $B$ on a number line such that the segment is partitioned in a ratio of $3: 4$. Point $A$ is at -6 and point $B$ is at 2. The student's work is shown.
1. $C=\left(\frac{3}{4}\right)(2-(-6))+(-6)$
2. $C=\left(\frac{3}{4}\right)(8)-6$
3. $c=6-6$
4. $C =0$
Analyze the student's work. Is the answer correct? Explain.
A. No, the student should have added $3+4$ to get the total number of sections, and used the fraction $\frac{3}{7}$ instead of $\frac{3}{4}$.
B. No, the student should have subtracted 2 from -6 to find the distance.
C. No, the student should have added 2 at the end to add to the starting point.
D. Yes, the student's answer is correct.
Answer (1)
The problem involves finding a point C on a line segment AB that divides it in the ratio 3:4.
Apply the section formula: C = A + m + n m ( B − A ) , where A = -6, B = 2, m = 3, and n = 4.
Substitute the values into the formula and simplify: C = − 6 + 7 3 ( 2 − ( − 6 )) = − 6 + 7 3 ( 8 ) = 7 − 18 .
The correct answer is 7 − 18 .
Explanation
Understanding the Problem We are given two points, A = − 6 and B = 2 , on a number line. We want to find a point C on the directed line segment from A to B such that the ratio of A C to CB is 3 : 4 . This means that the segment A C is 3 + 4 3 = 7 3 of the total length of the segment A B .
Applying the Section Formula To find the coordinate of point C , we can use the formula: C = A + m + n m ( B − A ) where A and B are the coordinates of the given points, and m : n is the given ratio. In this case, A = − 6 , B = 2 , m = 3 , and n = 4 .
Substituting the Values Substituting the given values into the formula, we get: C = − 6 + 3 + 4 3 ( 2 − ( − 6 ))
Calculating the Coordinate of C Simplifying the expression: C = − 6 + 7 3 ( 2 + 6 ) C = − 6 + 7 3 ( 8 ) C = − 6 + 7 24 C = 7 − 42 + 7 24 C = 7 − 18 C ≈ − 2.57
Analyzing the Student's Work Now let's analyze the student's work:
C = ( 4 3 ) ( 2 − ( − 6 )) + ( − 6 )
C = ( 4 3 ) ( 8 ) − 6
C = 6 − 6
C = 0 The student incorrectly used the fraction 4 3 instead of 7 3 in the formula. The student also made a calculation error in step 3. The correct calculation should be: C = 4 3 ( 8 ) − 6 = 6 − 6 = 0 . However, the initial setup was incorrect.
Conclusion The correct coordinate of point C is 7 − 18 ≈ − 2.57 , while the student's answer is 0 . Therefore, the student's answer is incorrect. The student should have used the fraction 7 3 instead of 4 3 .
Examples
Imagine you're designing a garden and want to place a bench along a path. The path goes from point A to point B, and you want the bench to be 7 3 of the way from A to B. By using the section formula, you can accurately determine the exact location to place the bench, ensuring it's proportionally positioned along the path. This concept is useful not only in garden design but also in urban planning, interior design, and any situation where proportional placement is important.
