The work of two students is shown for the problem: What is the slope of a line that is perpendicular to a line that passes through $(-21,12)$ and $(16,11)$?
| Student A | Student B |
| :--------------------------------- | :--------------------------------- |
| [tex]$m=\frac{11-12}{16-(-21)}=-\frac{1}{37}$[/tex] | [tex]$m=\frac{11-12}{16-(-21)}=-\frac{1}{37}$[/tex] |
| The slope of the perpendicular line is 37. | The slope of the perpendicular line is -37. |
Part A: Which student has the correct slope of the perpendicular line? (2 points)
Part B: Explain how the answer to Part A is correct. (2 points)
Answer (2)
Student A is correct; the slope of the line that is perpendicular to the line through points ( − 21 , 12 ) and ( 16 , 11 ) is 37. To find this, we first calculated the slope of the original line as − 37 1 and then found the negative reciprocal to determine the perpendicular slope. Hence, the correct answer is 37.
;
Calculate the slope of the line passing through the points ( − 21 , 12 ) and ( 16 , 11 ) using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 37 1 .
Determine the slope of the perpendicular line by taking the negative reciprocal of the original slope: m ⊥ = − m 1 = 37 .
Verify that the product of the original slope and the perpendicular slope is -1, confirming perpendicularity.
Conclude that Student A is correct, as the slope of the perpendicular line is 37 .
Explanation
Problem Analysis Let's first analyze the problem. We are given two points, ( − 21 , 12 ) and ( 16 , 11 ) , and we need to find the slope of a line perpendicular to the line passing through these points. Two students, A and B, have provided their solutions, and we need to determine which student is correct and explain why.
Calculate the slope The slope of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula:
m = x 2 − x 1 y 2 − y 1
In our case, ( x 1 , y 1 ) = ( − 21 , 12 ) and ( x 2 , y 2 ) = ( 16 , 11 ) . Plugging these values into the formula, we get:
m = 16 − ( − 21 ) 11 − 12 = 16 + 21 − 1 = 37 − 1
So, the slope of the line passing through the given points is − 37 1 .
Calculate the perpendicular slope The slope of a line perpendicular to a line with slope m is given by − m 1 . Therefore, the slope of the line perpendicular to the line with slope − 37 1 is:
m ⊥ = − − 37 1 1 = − ( − 37 ) = 37
Thus, the slope of the perpendicular line is 37.
Identify the correct student Comparing our result with the answers provided by the students, we see that Student A has the correct slope of the perpendicular line, which is 37.
Justify the answer Student A is correct because the product of the slopes of two perpendicular lines is -1. The slope of the original line is − 37 1 , and the slope of the perpendicular line is 37. Their product is:
( − 37 1 ) × 37 = − 1
This confirms that the lines are indeed perpendicular.
Final Answer Therefore, Student A has the correct slope of the perpendicular line, which is 37.
Examples
Understanding perpendicular slopes is crucial in many real-world applications. For example, architects use this concept to ensure walls are perfectly vertical to the ground, guaranteeing structural stability. Similarly, in navigation, knowing the perpendicular direction to a path helps in setting the correct course adjustments. In computer graphics, perpendicularity is used to calculate lighting effects on surfaces, creating realistic images.
