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(a) If a circle is inscribed in a right angled triangle ABC with the right angle at B, show the diameter of the circle is equal to AB + BC – AC. (b) If a triangle is inscribed in a circle, then the product of any two sides of the triangle is equal to the product of the diameter and the perpendicular distance of the third side from the opposite vertex. Prove the above statement.

(a) If a circle is inscribed in a right angled triangle ABC with the right angle at B, show the diameter of the circle is equal to AB + BC – AC.
(b) If a triangle is inscribed in a circle, then the product of any two sides of the triangle is equal to the product of the diameter and the perpendicular distance of the third side from the opposite vertex. Prove the above statement.

Grade:11

1 Answers

Aditi Chauhan
askIITians Faculty 396 Points
8 years ago
Hello Student,
Please find the answer to your question
(a) Inradius of the circle is given by
r = (s – b) tan B/2 = (a + b + c/2 – b) tan π/4 = a + c – b/2
2 r = a + c – b ⇒ Diameter = BC + AB – AC
(b) Given a ∆ABC in which AD ⊥ BC, AE is diameter of circumcircle of ∆ ABC.
236-2341_12345.png
To prove:
AB x AC = AE x AD
Construction: Join BE
Proof : ∠ABE = 90° (∠ in a semi circle)
Now in ∆’s ABE and ADC
∠ABE = ∠ADC (each 90°)
∠ AEB = ∠ACD (∠’s in the same segment)
∴ ∆ABE ~ ∆ADC (by AA similarity)
⇒ AB/AD = AE/AC
⇒ AB x AC x = AD x AR (Proved)

Thanks
Aditi Chauhan
askIITians Faculty

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