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# Prove that (4, 3), (6, 4) (5, 6) and (3, 5) Are the Angular Points of a Square. - CBSE Class 10 - Mathematics

ConceptConcepts of Coordinate Geometry

#### Question

Prove that (4, 3), (6, 4) (5, 6) and (3, 5)  are the angular points of a square.

#### Solution

Let A (4, 3); B (6, 4); C (5, 6) and D (3, 5) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a square.

So we should find the lengths of sides of quadrilateral ABCD.

AB = sqrt((6 - 4)^2 + (4 -3)^2)

= sqrt(4 + 1)

= sqrt5

BC = sqrt((6 - 5)^2 + (4 - 6)^2)

= sqrt(1 + 4)

= sqrt5

CD = sqrt((3 - 5)^2 + (5 - 6)^2)

= sqrt(4 + 1)

= sqrt5

AD = sqrt((3 - 4)^2 + (5 - 3)^2)

= sqrt(1+ 4)

= sqrt5

All the sides of quadrilateral are equal.

So now we will check the lengths of the diagonals.

AC = sqrt((5 - 4)^2 + (6 - 3)^2)

=sqrt(1 + 9)

= sqrt(10)

BC = sqrt((6 - 3)^2 + (4 - 5)^2)

= sqrt(9 + 1)

= sqrt10

All the sides as well as the diagonals are equal. Hence ABCD is a square.

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Solution Prove that (4, 3), (6, 4) (5, 6) and (3, 5) Are the Angular Points of a Square. Concept: Concepts of Coordinate Geometry.
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