Problem 1 :
In the diagram given below, find the slope of each line. Determine whether the lines j_{1 }and j_{2 }are perpendicular.
Problem 2 :
In the diagram given below, find the slope of each line. Determine whether the lines are perpendicular.
Problem 3 :
Decide whether the lines are perpendicular.
Line 1 : y = 3x/4 + 2
Line 2 : y = -4x/3 - 3
Problem 4 :
Decide whether the lines are perpendicular.
Line 1 : 4x + 5y = 2
Line 2 : 5x + 4y = 3
Problem 5 :
In the diagram given below, the equation y = 3x/2 + 3 represents a mirror. A ray of light hits the mirror at (-2, 0). What is the equation of the line p that is perpendicular to the mirror at this point ?
1. Answer :
Part 1 :
Find the slope of the line j_{1}. Line j_{1} is passing through the points (0, 3) and (3, 1).
Let (x_{1}, y_{1}) = (0, 3) and (x_{2}, y_{2}) = (3, 1).
Slope (j_{1}) = (y_{2} - y_{1})/(x_{2} - x_{1})
Slope (j_{1}) = (1 - 3)/(3 - 0)
Slope (j_{1}) = -2/3
Part 2 :
Find the slope of the line j_{2}. Line j_{2} is passing through the points (0, 3) and (-4, -3).
Let (x_{1}, y_{1}) = (0, 3) and (x_{2}, y_{2}) = (-4, -3).
Slope (j_{2}) = (-3 - 3)/(-4 - 0)
Slope (j_{2}) = -6/(-4)
Slope (j_{2}) = 3/2
Multiply the slopes :
The product is
= (-2/3) x (3/2)
= -1
Since the product of slopes of the lines j_{1 and }j_{2 }is -1, the lines j_{1 and }j_{2} are perpendicular.
2. Answer :
Part 1 :
Find the slope of the line AC. Line AC is passing through the points (1, -4) and (4, 2).
Let (x_{1}, y_{1}) = (1, -4) and (x_{2}, y_{2}) = (4, 2).
Slope of AC = (y_{2} - y_{1})/(x_{2} - x_{1})
= [(2 - (-4)]/(4 - 1)
= (2 + 4)/3
= 6/3
= 2
Part 2 :
Find the slope of the line BD. Line BD is passing through the points (-1, 2) and (5, -1).
Let (x_{1}, y_{1}) = (-1, 2) and (x_{2}, y_{2}) = (5, -1).
Slope of BD = (-1 - 2)/[(5 - (-1)]
= -3/6
= -1/2
Multiply the slopes :
The product is
= 2 x (-1/2)
= -1
Since the product of slopes of the lines is -1, the lines AC and BD are perpendicular.
3. Answer :
When we compare the given equations to slope intercept equation of a line y = mx + b, we get
slope of line 1 = 3/4
slope of line 2 = -4/3
Multiply the slopes :
The product is
= (3/4) x (-4/3)
= -1
Since the product of slopes of the lines is -1, the given lines are perpendicular.
4. Answer :
Rewrite each equation in slope-intercept form to find the slope.
Line 1 4x + 5y = 2 5y = -4x + 2 y = -4x/5 + 2/5 Slope = -4/5 |
Line 2 5x + 4y = 3 4y = -5x + 3 y = -5x/4 + 3/4 Slope = -5/4 |
Multiply the slopes :
The product is
= (-4/5) x (-5/4)
= 1
Since the product of slopes of the lines is not -1, the given lines are not perpendicular.
5. Answer :
The slope of the mirror is 3/2. So, the slope of the line p is -2/3.
Let y = mx + b be the equation of the line p.
Substitute (x, y) = (-2, 0) and m = -2/3 to find the value of b.
0 = (-2/3)(-2) + b
0 = 4/3 + b
Subtract 4/3 from both sides.
-4/3 = b
So, the equation of the line p is
y = -2x/3 - 4/3
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