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CBSE Class 9 Mathematics Question 15 of 20

Circles — Question 7

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Question 7

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

Answer

Let ABCD be a cyclic quadrilateral where diagonal AC and BD are diameters.

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle. NCERT Class 9 Mathematics CBSE Solutions.

Since BD is a diameter.

Arc BAD is a semicircle, So ∠BAD = 90° (Angle in a semi circle is a right angle)

Since AC ia a diameter.

Arc ABC is a semicircle, So ∠ABC = 90° (Angle in a semi circle is a right angle)

Also,

ABCD is a cyclic quadrilateral

From figure,

⇒ ∠BCD + ∠BAD = 180° (Sum of opposite angles of cyclic quadrilateral is 180°)

⇒ ∠BCD + 90° = 180°

⇒ ∠BCD = 180° - 90°

⇒ ∠BCD = 90°.

⇒ ∠ABC + ∠ADC =180° (Sum of opposite angles of cyclic quadrilateral is 180°)

⇒ ∠ADC + 90° = 180°

⇒ ∠ADC = 180° - 90°

⇒ ∠ADC = 90°.

So, in quadrilateral ABCD

∠A = ∠B = ∠C = ∠D = 90°

Since, all angles equal to 90°.

Hence, proved that ABCD is a rectangle.