Question

AC and BD Are chords of a circle which bisect each other prove that AC and BD are diameters and ABCD is a rectangle

Answer 1

Answer:

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Step-by-step explanation:

(i) It is given that AB and CD are the two chords of a circle

Let the point of intersection be O.

Join AB,BC,CD and AD.

In triangles AOB and COD,

∠AOB=∠COD ...(Vertically opposite angles)

OB=OD ....(O is the mid-point of BD)

OA=OC ....(O is the mid-point of AC)

△AOB≅△COD ....SAS test of congruence

∴AB=CD ....c.s.c.t.

Similarly, we can prove △AOD≅△BOC, then we get

AD=BC ....c.s.c.t.

So, □ABCD is a parallelogram, since opposite sides are equal in length.

So, opposite angles are equal as well.

So, ∠A=∠C

Also, for a cyclic quadrilateral opposite angles add up to 180

o

So, ∠A+∠C=180

o

∠A+∠A=180

o

∠A=90

o

So, BD is the diameter. Similarly, AC is also the diameter.

ii) Since AC and BD are diameters,

∴∠A=∠B=∠C=∠D=90

o

...Angle inscribed in a semi circle is a right angle

Hence, parallelogram ABCD is a rectangle