Ex 10.6, 7 (Optional)
AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters, (ii) ABCD is a rectangle.
Given: Chords AC and BD bisect each other
To Prove: (i) AC and BD are diameters
(ii) ABCD is a rectangle.
Proof:
In quadrilateral ABCD,
AC bisects BD (given)
So, diagonals bisect each other
∴ ABCD is a Parallelogram
(In a parallelogram, diagonals bisect each other)
Now,
Opposite angles of Parallelogram are equal
Now, ∠BAD = ∠BCD
and ∠ABC = ∠ADC
Also, ABCD is a cyclic quadrilateral,
∴ Sum of opposite angles is 180°
∠BAD + ∠BCD = 180°
∠BCD + ∠BCD = 180°
2 ∠BCD = 180°
∠ BCD = (180°)/2
∠ BCD = 90°
We know that
Diameter subtends 90° angle on circle
∴ BD must be diameter
Similarly,
We can prove that
AC must be diameter
Now,
ABCD is a parallelogram with one angle 90°
∴ ABCD is a rectangle
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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