a parallelogram has vertices (0,2), (2,-1),(6,3),(p,q). which of the following ordered pair has possible values for (p,q)
a) (0,6)
b) (6,1)
c) (4,6)
d) (6,4)
Plot the given vertices on a graph
and im sure you can find which is the last one
thanks so much...its so easy and i've been thinking about it since the afternoon. thanks!!!!!!
To determine the possible values for (p,q) in the parallelogram, we can use the fact that opposite sides of a parallelogram are equal in length and parallel.
Using the given vertices, we can find the lengths of the sides of the parallelogram:
Side 1: Length between (0,2) and (2,-1)
Length = √((2-0)^2 + (-1-2)^2) = √(2^2 + (-3)^2) = √(4+9) = √13
Side 2: Length between (2,-1) and (6,3)
Length = √((6-2)^2 + (3-(-1))^2) = √(4^2 + 4^2) = √(16+16) = √32 = 4√2
We need to find another side with the same length as sides 1 and 2.
Let's consider option (a) (0,6) as a possible value for (p,q):
Side 3: Length between (6,3) and (0,6)
Length = √((0-6)^2 + (6-3)^2) = √((-6)^2 + 3^2) = √(36+9) = √45 = 3√5
Since the length of Side 3 is not equal to the lengths of sides 1 and 2, the ordered pair (0,6) does not have the correct values for (p,q).
Now let's consider option (b) (6,1) as a possible value for (p,q):
Side 3: Length between (6,3) and (6,1)
Length = √((6-6)^2 + (1-3)^2) = √(0^2 + (-2)^2) = √(0+4) = √4 = 2
Since the length of Side 3 is not equal to the lengths of sides 1 and 2, the ordered pair (6,1) does not have the correct values for (p,q).
Next, let's consider option (c) (4,6) as a possible value for (p,q):
Side 3: Length between (6,3) and (4,6)
Length = √((4-6)^2 + (6-3)^2) = √((-2)^2 + 3^2) = √(4+9) = √13
Since the length of Side 3 is equal to the lengths of sides 1 and 2, the ordered pair (4,6) has the correct values for (p,q).
Finally, let's consider option (d) (6,4) as a possible value for (p,q):
Side 3: Length between (6,3) and (6,4)
Length = √((6-6)^2 + (4-3)^2) = √(0^2 + 1^2) = √1 = 1
Since the length of Side 3 is not equal to the lengths of sides 1 and 2, the ordered pair (6,4) does not have the correct values for (p,q).
Therefore, the only ordered pair with possible values for (p,q) is (c) (4,6).
To determine the possible values for (p, q) in the given parallelogram, we need to use the properties of a parallelogram.
One property of a parallelogram is that opposite sides are parallel. This means that the slope of one side should be equal to the slope of the other side.
Let's consider the sides of the parallelogram formed by the given vertices:
Side AB: (0, 2) to (2, -1)
Side BC: (2, -1) to (6, 3)
Side CD: (6, 3) to (p, q)
Side AD: (p, q) to (0, 2)
First, let's find the slopes of AB and BC:
Slope of AB = (change in y)/(change in x) = (-1 - 2)/(2 - 0) = -3/2
Slope of BC = (change in y)/(change in x) = (3 - (-1))/(6 - 2) = 4/4 = 1
Since opposite sides of a parallelogram are parallel, the slopes of AB and CD should also be equal.
Now let's consider the slopes of CD and AD:
Slope of CD = (change in y)/(change in x) = (q - 3)/(p - 6)
Slope of AD = (change in y)/(change in x) = (2 - q)/(0 - p) = (q - 2)/(p - 0) = (q - 2)/p
Setting the slopes of CD and AD equal to the slope of AB, we get:
(q - 3)/(p - 6) = -3/2
Now, we can solve this equation to find the possible values for (p, q).
Cross-multiplying the equation, we get:
2(q - 3) = -3(p - 6)
2q - 6 = -3p + 18
2q + 3p = 24 (rearranging the terms)
Now, let's substitute the given options for (p, q) and check if they satisfy the equation:
a) (0, 6)
2(6) + 3(0) = 12 + 0 = 12 ≠ 24, not a solution
b) (6, 1)
2(1) + 3(6) = 2 + 18 = 20 ≠ 24, not a solution
c) (4, 6)
2(6) + 3(4) = 12 + 12 = 24, a solution
d) (6, 4)
2(4) + 3(6) = 8 + 18 = 26 ≠ 24, not a solution
Therefore, the possible value for (p, q) is (4, 6), so the answer is option c) (4, 6).