The two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of the other two vertices
Solution:
Let's draw a figure of a square with the two opposite vertices (-1, 2) and (3, 2),
Let ABCD be a square having known vertices A (- 1, 2) and C (3, 2) respectively.
Let B(x₁, y₁) and D(x₂, y₂) be the two unknown vertex
We know that the sides of a square are equal to each other.
Therefore, AB = BC
By Using Distance formula on AB = AC with A (- 1, 2), B(x₁, y₁) and C (3, 2)
√ [(x₁ - (-1))2 + (y₁ - 2)2] = √ [(x₁ - 3)2 + (y₁ - 2)2]
x₁2 + 2x₁ + 1 + y₁2 - 4y₁ + 4 = x₁2 + 9 - 6x₁ + y₁2 + 4 - 4y₁ (By Simplifying & Transposing)
8x₁ = 8
x₁ = 1
We know that in a square, all interior angles are 90 degrees.
In ΔABC
AB2 + BC2 = AC2 [By Pythagoras theorem]
The distance formula is used to find the distance between AB, BC, and AC
(x₁ - (-1))2 + (y₁ - 2)2 + (x₁ - 3)2 + (y₁ - 2)2 = [3 - (-1)]2 + [ 2 - 2 ]2
By using x₁ = 1,
(1 + 1)2 + (y₁ - 2)2 + (1 - 3)2 + (y₁ - 2)2 = 16
4 + y₁2 + 4 - 4y₁ + 4 + y₁2 - 4y₁ + 4 = 16
2y₁2 + 16 - 8y₁ = 16
2y₁2 - 8y₁ = 0
y₁ (y₁ - 4) = 0
y₁ = 0 or 4
Now, we have got the coordinates of point B(1, 0)
Let's plot the square on a graph as shown below:
We see that the vertex opposite to (1, 0) is (1, 4)
Hence, for point D we have the coordinates x₂ = 1, y₂ = 4
Hence the required vertices are B (1, 0) and D (1, 4).
☛ Check: NCERT Solutions Class 10 Maths Chapter 7
Video Solution:
The two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of the other two vertices.
Maths NCERT Solutions Class 10 Chapter 7 Exercise 7.4 Question 4
Summary:
The two opposite vertices of a square are (- 1, 2) and (3, 2). Then the coordinates of the other two vertices are B (1, 0) and D (1, 4).
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