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The two opposite vertices of a square are (− 1, 2) and (3, 2). Find the coordinates of the other two vertices. - CBSE Class 10 - Mathematics

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Question

The two opposite vertices of a square are (− 1, 2) and (3, 2). Find the coordinates of the other two vertices.

Solution

Let ABCD be a square having (−1, 2) and (3, 2) as vertices A and C respectively. Let (xy), (x1y1) be the coordinate of vertex B and D respectively.

We know that the sides of a square are equal to each other.

∴ AB = BC

`=>sqrt((x+1)^2 + (y-2)^2) = sqrt((x-3)^2 + (y-2)^2)`

=>x2 + 2x + 1 + y2 -4y + 4 = x2 + 9  -6x + y2 + 4 - 4y

⇒ 8x = 8

⇒ x = 1

We know that in a square, all interior angles are of 90°.

In ΔABC,

AB2 + BC2 = AC2

`=> (sqrt(((1+1)^2)+(y-2)^2))^2 + (sqrt(((1-3)^2)+(y-2)^2))^2 = (sqrt((3+1)^2+(2-2)^2))^2`

⇒ 4 + y2 + 4 − 4y + 4 + y2 − 4y + 4 =16

⇒ 2y2 + 16 − 8 y =16

⇒ 2y2 − 8 y = 0

⇒ y (− 4) = 0

⇒ y = 0 or 4

We know that in a square, the diagonals are of equal length and bisect each other at 90°. Let O be the mid-point of AC. Therefore, it will also be the mid-point of BD

Coordinate of point O = ((-1+3)/2, (2+2)/2)

`((1+x_1)/2, (y+ y_1)/2) = (1,2)`

`(1+x_1)/2 = 1`

1+x1=2

x1 =1

and

` (y + y_1)/2 = 2`

⇒ y + y1 = 4

If y = 0,

y1 = 4

If y = 4,

y1 = 0

Therefore, the required coordinates are (1, 0) and (1, 4).

  Is there an error in this question or solution?

APPEARS IN

 NCERT Solution for Mathematics Textbook for Class 10 (2019 to Current)
Chapter 7: Coordinate Geometry
Ex. 7.40 | Q: 4 | Page no. 171

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Solution The two opposite vertices of a square are (− 1, 2) and (3, 2). Find the coordinates of the other two vertices. Concept: Area of a Triangle.
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