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

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Answer 2

Answer:

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