AC and BD are chord of a circle which bisect each other prove that AC and BD are the diameters and ABCD is a rectangle
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hey I am vipul here
Let AC and BD are the chords of a circle which bisect each other.
1. We have to prove that AC and BD are diameter
Since it is given that AC and BD are the chords of a circle which bisect each other.
Now in ΔABD,
∠A = 90
So BD is a diameter (since angle in a semi-circle is 90)
Again in ΔBCD,
∠D = 90
So AC is a diameter (since angle in a semi-circle is 90)
So AC and BD are the diameters.
2. We have to prove that ABCD is a rectangle
Let AC and BD are the chords of a circle which bisect each other at the point O.
Now in ΔOAB and ΔOCD
OA = OC (given)
OB = OD (given)
∠AOB = ∠COD (since vertically opposite angles)
So by SAS congruent criteria,
ΔOAB ≅ ΔOCD
BY CPCT
AB = CD ...........1
Similarly we can show that
AD = CB ..........2
Add equation 1 and 2, we get
AB + AD = CD + CB
=> ∠BAD = ∠BCD
So BD divides the circle into two equal semi-circle and the angle of each one is 90
So ∠A = 90 and ∠C = 90 ............3
Similarly we can show that
∠B = 90 and ∠D = 90 .........4
From equation 3 and 4, we get
∠A = ∠B =∠C = ∠D = 90
So ABCD is a rectangle.
Thanxx
Let AC and BD are the chords of a circle which bisect each other.
1. We have to prove that AC and BD are diameter
Since it is given that AC and BD are the chords of a circle which bisect each other.
Now in ΔABD,
∠A = 90
So BD is a diameter (since angle in a semi-circle is 90)
Again in ΔBCD,
∠D = 90
So AC is a diameter (since angle in a semi-circle is 90)
So AC and BD are the diameters.
2. We have to prove that ABCD is a rectangle
Let AC and BD are the chords of a circle which bisect each other at the point O.
Now in ΔOAB and ΔOCD
OA = OC (given)
OB = OD (given)
∠AOB = ∠COD (since vertically opposite angles)
So by SAS congruent criteria,
ΔOAB ≅ ΔOCD
BY CPCT
AB = CD ...........1
Similarly we can show that
AD = CB ..........2
Add equation 1 and 2, we get
AB + AD = CD + CB
=> ∠BAD = ∠BCD
So BD divides the circle into two equal semi-circle and the angle of each one is 90
So ∠A = 90 and ∠C = 90 ............3
Similarly we can show that
∠B = 90 and ∠D = 90 .........4
From equation 3 and 4, we get
∠A = ∠B =∠C = ∠D = 90
So ABCD is a rectangle.
Thanxx
ali135:
thanks
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