Assessment 6
1. The coordinates of line [tex]$AB$[/tex] are [tex]$A(5,4)$[/tex] and [tex]$B(-2, t)$[/tex]. The line is perpendicular to a line whose gradient is [tex]$-\frac{1}{2}$[/tex]. Determine the value of [tex]$t$[/tex].
2. The coordinates of line [tex]$AB$[/tex] are [tex]$A(3,6)$[/tex] and [tex]$B(1,1)$[/tex]. Determine the gradient of a line perpendicular to line [tex]$AB$[/tex].
3. Without drawing, determine whether each of the following pairs of lines are perpendicular or not.
(a) [tex]$y=-3 x+4$[/tex] and [tex]$y=\frac{1}{3} x+12$[/tex]
(b) [tex]$y=\frac{3}{5} x+3$[/tex] and [tex]$y=5 x-12$[/tex]
(c) [tex]$y=\frac{1}{7} x-9$[/tex] and [tex]$y=-\frac{1}{7} x+11$[/tex]
4. Chebet drew line [tex]$NM$[/tex] with coordinates [tex]$N(3,5)$[/tex] and [tex]$M(1,-1)$[/tex]. Magera drew line [tex]$CD$[/tex] with coordinates [tex]$C(3,6)$[/tex] and [tex]$D(4,3)$[/tex]. Determine whether line [tex]$NM$[/tex] is perpendicular to line [tex]$CD$[/tex] or not.
Answer (2)
Find t using the perpendicular gradient relationship: t = − 10 .
Determine the perpendicular gradient: − 5 2 .
Check perpendicularity by multiplying gradients: (a) Perpendicular, (b) Not Perpendicular, (c) Not Perpendicular.
Verify if lines NM and C D are perpendicular: Not Perpendicular. See details in the step-by-step solution.
Explanation
Problem Analysis We are given the coordinates of line A B as A ( 5 , 4 ) and B ( − 2 , t ) . We are also given that line A B is perpendicular to a line with gradient − 2 1 . Our goal is to find the value of t .
Calculate the gradient of line AB The gradient of line A B , denoted as m A B , can be calculated using the coordinates of points A and B :
m A B = − 2 − 5 t − 4 = − 7 t − 4 Since line A B is perpendicular to a line with gradient − 2 1 , the product of their gradients must be − 1 . Therefore: m A B × ( − 2 1 ) = − 1
Solve for m_AB Now, we solve for m A B :
m A B = − 2 1 − 1 = 2 Substitute m A B = 2 into the equation for the gradient of line A B :
− 7 t − 4 = 2
Solve for t Solve for t :
t − 4 = 2 × ( − 7 ) t − 4 = − 14 t = − 14 + 4 t = − 10
Find the value of t Therefore, the value of t is − 10 .
Next, we are given the coordinates of line A B as A ( 3 , 6 ) and B ( 1 , 1 ) . We need to find the gradient of a line perpendicular to line A B .
Calculate the gradient of a line perpendicular to AB First, calculate the gradient of line A B :
m A B = 1 − 3 1 − 6 = − 2 − 5 = 2 5 The gradient of a line perpendicular to A B , denoted as m ⊥ , is the negative reciprocal of m A B :
m ⊥ = − m A B 1 = − 2 5 1 = − 5 2
Determine if pairs of lines are perpendicular Therefore, the gradient of a line perpendicular to line A B is − 5 2 .
Now, we need to determine whether each of the following pairs of lines are perpendicular or not: (a) y = − 3 x + 4 and y = 3 1 x + 12 (b) y = 5 3 x + 3 and y = 5 x − 12 (c) y = 7 1 x − 9 and y = − 7 1 x + 11
Check each pair of lines (a) The gradients of the lines are m 1 = − 3 and m 2 = 3 1 . The product of the gradients is: m 1 × m 2 = − 3 × 3 1 = − 1 Since the product is − 1 , the lines are perpendicular.
(b) The gradients of the lines are m 1 = 5 3 and m 2 = 5 . The product of the gradients is: m 1 × m 2 = 5 3 × 5 = 3 Since the product is not − 1 , the lines are not perpendicular.
(c) The gradients of the lines are m 1 = 7 1 and m 2 = − 7 1 . The product of the gradients is: m 1 × m 2 = 7 1 × ( − 7 1 ) = − 49 1 Since the product is not − 1 , the lines are not perpendicular.
Determine if lines NM and CD are perpendicular Finally, Chebet drew line NM with coordinates N ( 3 , 5 ) and M ( 1 , − 1 ) . Magera drew line C D with coordinates C ( 3 , 6 ) and D ( 4 , 3 ) . We need to determine whether line NM is perpendicular to line C D or not.
Calculate gradients and check for perpendicularity Calculate the gradient of line NM :
m NM = 1 − 3 − 1 − 5 = − 2 − 6 = 3 Calculate the gradient of line C D :
m C D = 4 − 3 3 − 6 = 1 − 3 = − 3 Check if the lines are perpendicular by multiplying their gradients: m NM × m C D = 3 × ( − 3 ) = − 9 Since the product is not − 1 , the lines are not perpendicular.
Final Answers
The value of t is − 10 .
The gradient of a line perpendicular to line A B is − 5 2 .
(a) The lines are perpendicular .
(b) The lines are not perpendicular .
(c) The lines are not perpendicular .
Line NM is not perpendicular to line C D .
Examples
Understanding perpendicular lines is crucial in many real-world applications. For example, architects use perpendicular lines to design buildings, ensuring walls meet floors at right angles for stability. In navigation, understanding perpendicular vectors helps in calculating the shortest distance between two points when considering factors like wind or current. Additionally, in computer graphics, perpendicularity is essential for creating realistic lighting and shadows, enhancing the visual depth and accuracy of rendered images.
The value of t is -10. The gradient of a line perpendicular to line AB is -\frac{2}{5}. The pairs of lines with respective relationships are: (a) perpendicular, (b) not perpendicular, (c) not perpendicular, and lines NM and CD are not perpendicular.
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