Question

If

A B C D

is a square then show that the points

A, B, C,

and

D

are concyclic.

Accepted Answer

Consider a square $A B C D$ of side length $a$ units whose one of the vertex lies at the origin and the other two vertices lies on the $x$ -axis and $y$ -axis such that $A = (a, 0), B = (a, a), C = (0, a),$ and $D = (0, 0)$

We know that three non-collinear points always form a triangle through which a circle can be drawn which is known as a circumcircle and its centre is known as a circumcentre.

Since $A, B,$ and $C$ are the vertices of a square and non-collinear points, so they form a circle.

And, $∆ A B C$ is a right-angled triangle and the circumcentre of a right-angled triangle lies exactly the midpoint of the hypotenuse.

$∴$ Circumcentre of the circle formed by the vertices $A, B, C =$ Mid point of hypotenuse $A C$

$= (\frac{a + 0}{2}, \frac{0 + a}{2}) = (\frac{a}{2}, \frac{a}{2})$

Similarly, if we consider the vertices $A, D, and B$ then, $∆ A D B$ is also a right angled triangle.

$∴$ Circumcentre of the circle formed by the vertices $A, D,$ and $B =$ Midpoint of $B D$

$= (\frac{a}{2}, \frac{a}{2})$

Since the midpoints of hypotenuse of $∆ A B C$ and $∆ A D B$ coincides, it means the vertices $A, B, C,$ and $D$ are the points on the same circle.

Hence, If $A B C D$ is a square then the points $A, B, C,$ and $D$ are concyclic.

Andhra Pradesh Board Solutions for Exercise 1: Exercise 1(a)

Andhra Pradesh Board Mathematics Solutions for Exercise - Andhra Pradesh Board Solutions for Exercise 1: Exercise 1(a)

Questions from Andhra Pradesh Board Solutions for Exercise 1: Exercise 1(a) with Hints & Solutions

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