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Ex.10.5 Q7 Circles Solution - NCERT Maths Class 9

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Question

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

 Video Solution
Circles
Ex 10.5 | Question 7

Text Solution

What is known?

Diagonals of a cyclic quadrilateral are diameters of a circle passing through the vertices.

What is unknown?

To prove lines joining the vertices is a rectangle.

Reasoning:

  • The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
  • The sum of either pair of opposite angles of a cyclic quadrilateral is \(180^{\circ}.\)
  • Diameter is a chord.

Steps:

Let \({DB}\) be the diameter of the circle which is also a chord.

Then  \(\begin {align} \angle {BOD}=180^{\circ} \end {align}\)

We know that, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

\[\begin{align} \angle {BAD} &=\frac{1}{2} \times \angle {BOD} \\ &=90^{\circ} \end{align}\]

Similarly,  \(\begin {align} \angle {BCD}=90^{\circ} \end {align}\)

Now considering \(\text{AC}\) as the diameter of the circle, we get \(\begin {align} \angle {AOC}=180^{\circ} \end {align}\)

We know that, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

\[\begin{align} \angle {ABC} &=\frac{1}{2} \times \angle {AOC} \\ &=90^{\circ} \end{align}\]

Similarly, \(\begin {align} \angle {ADC}=90^{\circ} \end {align}\)

As you can see, all the angles at the corners are \(90^\circ \)we can say that the shape joining the vertices is a rectangle.

This problem can also be solved by using the property of cyclic quadrilaterals.

 Video Solution
Circles
Ex 10.5 | Question 7
  
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