If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
Solution:
Concepts used to solve the question:
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The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
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The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
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Diameter is the longest chord.
Let BD be the diameter of the circle, which is also a chord. Then, ∠BOD = 180°
Since AC and BD are diameters of the circle
⇒ AC = BD
We know that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
∴ ∠BAD = 1/2 × ∠BOD = 90°
Similarly, ∠BCD = 90°
Now, considering AC as the diameter of the circle, we get ∠AOC = 180°
We know that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
∠ABC = 1/2 × ∠AOC = 90°
Similarly, ∠ADC = 90°
Let us now consider the triangles ∆ABC and ∆BAD,
∠ABC = ∠BAD [Each equal to 90°]
AB = BA [Common side of triangle ABC and BAD]
AC = BD [Diameter of the circle]
∴ ∆ABC ≅ ∆BAD [By RHS congruence criteria]
⇒ BC = AD [By Corresponding parts of Congruent triangles theorem]
Similarly, AB = DC
As you can see, all the angles at the corners are 90º and the opposite sides are equal. We can say that the shape joining the vertices ABCD is a rectangle.
☛ Check: NCERT Solutions for Class 9 Maths Chapter 10
Video Solution:
If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
Maths NCERT Solutions Class 9 Chapter 10 Exercise 10.5 Question 7
Summary:
If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, then it is a rectangle.
☛ Related Questions:
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