Quadrilateral ABCD has vertices A(-2,1), B(3,3), C(5,7), and D(0,5). Determine whether line AC is perpendicular to line BD.
Is line AC perpendicular to line BD
Two lines are perpendicular if their slopes are negative reciprocals of each other (m1 * m2 = -1).
I'll leave the work for finding these lines to you.
AC--through (-2,1), (5,7)
y = 6/7 x + 19/7
slope m1 = 6/7
BD--through (3,3), (0,5)
y = -2/3 x + 5
slope m2 = -2/3
If perpendicular,
m1 * m2 = -1
6/7 * -2/3 = ?
Are the lines perpendicular?
To determine whether line AC is perpendicular to line BD, we need to calculate the slopes of these two lines and check if their product is -1.
Step 1: Calculate the slope of line AC:
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
For line AC:
Coordinates of point A: A(-2, 1)
Coordinates of point C: C(5, 7)
slope_AC = (7 - 1) / (5 - (-2))
= 6 / 7
Step 2: Calculate the slope of line BD:
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
For line BD:
Coordinates of point B: B(3, 3)
Coordinates of point D: D(0, 5)
slope_BD = (5 - 3) / (0 - 3)
= -2 / -3
= 2 / 3
Step 3: Check if the product of the slopes is -1:
Multiply the slopes of line AC and line BD:
slope_AC * slope_BD = (6 / 7) * (2 / 3)
= 12 / 21
= 4 / 7
Since the product of the slopes is not -1, line AC is not perpendicular to line BD.
Therefore, line AC is not perpendicular to line BD.