On a number line, the directed line segment from [tex]$Q$[/tex] to [tex]$S$[/tex] has endpoints [tex]$Q$[/tex] at -8 and [tex]$S$[/tex] at 12. Point [tex]$R$[/tex] partitions the directed line segment from [tex]$Q$[/tex] to [tex]$S$[/tex] in a 4:1 ratio. Which expression correctly uses the formula [tex]$\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$[/tex] to find the location of point R?
A. [tex]$\left(\frac{1}{1+4}\right)(12-(-8))+(-8)$[/tex]
B. [tex]$\left(\frac{4}{4+1}\right)(12-(-8))+(-8)$[/tex]
C. [tex]$\left(\frac{4}{4+1}\right)(-8-12)+12$[/tex]
D. [tex]$\left(\frac{4}{1+4}\right)(-8-12)+12$[/tex]
Answer (2)
Identify the coordinates of the endpoints: x 1 = − 8 and x 2 = 12 .
Determine the ratio m : n = 4 : 1 , so m = 4 and n = 1 .
Substitute the values into the section formula: ( m + n m ) ( x 2 − x 1 ) + x 1 = ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) .
The correct expression is: ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) .
Explanation
Problem Analysis Let's analyze the problem. We are given a line segment QS on a number line, with Q at -8 and S at 12. Point R divides this segment in a 4:1 ratio. We need to find the correct expression using the section formula to determine the location of point R. The section formula is given by ( m + n m ) ( x 2 − x 1 ) + x 1 , where x 1 and x 2 are the coordinates of the endpoints, and m : n is the ratio in which the point divides the segment.
Applying the Section Formula In our case, x 1 = − 8 (coordinate of Q) and x 2 = 12 (coordinate of S). The ratio is m : n = 4 : 1 , so m = 4 and n = 1 . Now, we substitute these values into the section formula: ( m + n m ) ( x 2 − x 1 ) + x 1 = ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) = ( 5 4 ) ( 12 + 8 ) − 8 = ( 5 4 ) ( 20 ) − 8
Comparing with the given options Now let's compare the expression we derived with the given options:
Option 1: ( 1 + 4 1 ) ( 12 − ( − 8 )) + ( − 8 ) - Incorrect, as the ratio is inverted. Option 2: ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) - Correct, matches our derived expression. Option 3: ( 4 + 1 4 ) ( − 8 − 12 ) + 12 - Incorrect, as the coordinates and signs are mixed up. Option 4: ( 1 + 4 4 ) ( − 8 − 12 ) + 12 - Incorrect, as the ratio is inverted and coordinates/signs are mixed up.
Final Answer Therefore, the correct expression is ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) .
Examples
The section formula is useful in various real-world scenarios, such as determining the location of a point on a map given its distance from two known locations, or in computer graphics for interpolating colors or positions between two points. For instance, if you're designing a game and want an object to move a certain fraction of the way between two points, you can use the section formula to calculate the object's new position at any given time. This ensures smooth and proportional movement, enhancing the user experience.
Using the section formula, the correct expression to find the location of point R that divides the segment from Q to S in a 4:1 ratio is option B: ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) . This correctly applies the partition ratio and coordinates of the endpoints.
;
