To find the missing coordinate of a parallelogram, we use one of the following methods.
(i) Using slope
(ii) Using midpoint formula
(iii) Using section formula
How to find the missing coordinate of a parallelogram using slope ?
In a parallelogram, the opposite sides are parallel. If two lines are parallel, then its slopes will be equal.
Let us consider the parallelogram ABCD,
Slope of AB = Slope of CD
Slope of BC = Slope of AD
How to find the missing coordinate of a parallelogram using midpoint formula ?
In a parallelogram, the diagonals will bisect each other.
That is, the midpoint of one diagonal is also the midpoint of another diagonal.
Let us consider the parallelogram ABCD,
Midpoint of AC = Midpoint of BD
How to find missing coordinate of parallelogram using section formula ?
The point of intersection of diagonals divides the diagonals in the same ratio.
For example, if the the common point of both diagonals is in the ratio l : m, then the value of l and m are equal.
Let us look at some examples to understand the above explained concepts.
Example 1 :
If (7, 3), (6, 1), (8, 2) and (p, 4) are the vertices of a parallelogram taken in order, then find the value of p.
Solution :
Let the vertices of the parallelogram be A (7, 3), B(6, 1), C (8, 2) and D (p, 4)
We know that the diagonals of a parallelogram bisect each other. The midpoints of the diagonal AC and the diagonal BD coincide.
Midpoint of AC = Midpoint of BD
Midpoint = (x1 + x2)/2 , (y1 + y2)/2
A (7, 3) and C (8, 2)
= (7 + 8)/2 , (3 + 2)/2
= (15/2, 5/2) ---------(1)
B (6, 1) and D (p, 4)
= (6 + p)/2 , (1 + 4)/2
= (6 + p)/2 , 5/2 ---------(2)
(15/2, 5/2) = ((6 + p)/2 , 5/2)
Equating x-coordinates, we get
15/2 = (6 + P) / 2
15 = 6 + p
Subtract 6 on both sides,
15 - 6 = 6 + p - 6
9 = p
So, the missing coordinate is 9.
Example 2 :
Using the concept of slope, find the missing coordinate (-2 , -1), (4 , 0), (a , 3) and (-3 , 2) taken in order form a parallelogram.
Solution :
Let the vertices of the parallelogram be A (-2 , -1), B (4 , 0), C (a , 3) and D (-3 , 2)
Slope of AB = Slope of CD
Slope = (y2- y1) / (x2-x1)
Slope of AB :
A (-2 , -1), B (4 , 0)
m = (0 - (-1)) / (4 - (-2))
m = (0 + 1) / (4 + 2)
m = 1 / 6 ------(1)
Slope of CD :
C (a , 3) and D (-3 , 2)
m = (2 - 3) / (-3 - a)
m = - 1 / (-3 - a)
m = 1 / (3 + a) ------(2)
1/6 = 1/(3 + a)
3 + a = 6
Subtract 3 on both sides
3 + a - 3 = 6 - 3
a = 3
So, the missing coordinate is 3.
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