Misc 1
Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). Find the coordinates of the fourth vertex.
In parallelogram, diagonals bisect each other
Hence
AO = OC
& BO = OD
We can say that
O is Midpoint of AC
and
O is Midpoint of BD
Finding co-ordinates of O
Now, O is the mid-point of AC
Here, A (3, –1. 2 ) & C ( –1 , 1, 2)
We know that
If O (x, y, z) is the mid point of A (x1, y1, z1) & B (x2, y2, z2) , then coordinates of O
O (x, y, z) = ((〖 𝑥〗_1 +〖 𝑥〗_2)/2,(〖 𝑦〗_1 + 〖 𝑦〗_2)/2,(𝑧_1 +〖 𝑧〗_2)/2)
Here,
x1 = 3, y1 = –1, z1 = 2
x2 = –1 , y2 = 1, z2 = 2
Putting values,
O = ((3 + (−1))/2, (−1 + 1)/2, (2 + 2)/2)
= ((3 − 1)/2, 0/2, 4/2)
= (2/2, 0, 2)
= (1, 0, 2)
Thus, the co-ordinates of O is (1, 0, 2)
Now,
O is also the mid-point of BD ,
Finding coordinates of O,
Here, B (1, 2, –4) , O (1, 0, 2)
Let D Be (x, y, z)
Now
Co-ordinates of O = ((1 + 𝑥)/2,(2 + 𝑦)/2, (−4 + 𝑧)/2)
(1, 0, 2) = ((1 + 𝑥)/2, (2 + 𝑦)/2, (−4 + 𝑧)/2)
Thus, x = 1 , y = -2 , z = 8
∴ Coordinates of point D are (1, –2, 8)
Thus, Coordinates of the fourth vertex are (1, –2, 8)
(1 + 𝑥)/2=1
1 + x = 2
x = 2 – 1
x = 1
(2 + 𝑦)/2=0
2 + y = 0 × 2
2 + y = 0
y = – 2
(−4 + 𝑧)/2=2
– 4 + z = 2 × 2
z = 4 + 4
z = 8

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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