(2, 3) and (4, 1) arethe two opposite vertices of a square. the area of the square is

The two opposite vertices of a square are 1,2 and 3,2. Find the co ordinates of the other two vertices.

B (1, 0) or B ( 1,4) and D(2, 4) or D(2,0)

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B (2,3) or B ( 2,4) and D(3, 4) or D(3,0)

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B (2,0) or B ( 1,4) and D(2, 4) or D(1,0)

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B (0,1) or B (0,2) and D(3, 1) or D(3,-2)

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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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