The exterior angles B and C in - ABC are bisected to meet at a point P- i Is - BPC-90o- A-2-ii Is ABPC a cyclic quadrilateral -

The correct option is B (i) Yes . (ii) No. nbsp;Given Δ ABC has its external angles ∠ CBR & ∠ BCQ bisected by the lines ...

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Byju's Answer

Standard IX

Mathematics

Vertically Opposite Angles

The exterior ...

Question

The exterior angles B and C in $Δ A B C$ are bisected to meet at a point P. (i) Is $∠ B P C = 90^{o} - \frac{A}{2}$ ?
(ii) Is ABPC a cyclic quadrilateral ?

(i) No. (ii) Yes

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(i) Yes . (ii) No.

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(i) No (ii) Can't be determined.

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(i) Can't be determined. (ii) No.

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Solution

The correct option is B (i) Yes . (ii) No.
$G i v e n - Δ A B C h a s i t s e x t e r n a l a n g l e s ∠ C B R & ∠ B C Q b i s e c t e d b y t h e l i n e s B P & C P r e s p e c t i v e l y . B P & C P i n t e r s e c t a t P . T o f i n d o u t - (i) i s ∠ B P C = 90^{o} - \frac{A}{2} ? (i i) i s A B P C a c y c l i c q u a d r i l a t e r a l ? S o l u t i o n - ∠ P B C = ∠ P B R ⟹ ∠ P B C + ∠ P B R = 2 ∠ P B C . S o ∠ A B C + 2 ∠ P B C = 180^{o} (l i n e a r p a i r) . i . e ∠ P B C = 90^{o} - \frac{∠ A B C}{2} . . . . . . . . (i) . S i m i l a r l y ∠ P C Q = ∠ P C B ⟹ ∠ P C B + ∠ P C Q = 2 ∠ P C B . S o ∠ A C B + 2 ∠ P C B = 180^{o} (l i n e a r p a i r) . i . e ∠ P C B = 90^{o} - \frac{∠ A C B}{2} . . . . . . . . (i i) . A d d i n g (i) & (i i) w e h a v e ∠ P B C + ∠ P C B = 90^{o} - \frac{∠ A B C}{2} + 90^{o} - \frac{∠ A C B}{2} = 180^{o} - \frac{1}{2} (∠ A B C + ∠ A C B) ⟹ 180^{o} - ∠ B P C = 180^{o} - \frac{1}{2} {180^{o} - ∠ B A C} ⟹ ∠ B P C = 90^{o} - \frac{B A C}{2} . . . . . . . (a n s - i) . A g a i n t h e s u m o f t h e o p p o s i t e a n g l e s o f t h e q u a d r i l a t e r a l A B P C = ∠ B A C + ∠ B P C = ∠ B A C + 90^{o} - \frac{B A C}{2} \neq 180^{o} . ∴ T h e q u a d r i l a t e r a l A B P C i s n o t a c y c l i c q u a d r i l a t e r a l . . . . . . . . . (a n s - i i) . (∵ t h e s u m o f t h e o p p o s i t e a n g l e s o f a c y c l i c q u a d r i l a t e r a l = 180^{o}) . A n s - O p t i o n B .$

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The exterior angles B and C in ΔABC are bisected to meet at a point P. (i) Is ∠BPC=90o−A2?(ii) Is ABPC a cyclic quadrilateral ?

The exterior angles B and C in $Δ A B C$ are bisected to meet at a point P. (i) Is $∠ B P C = 90^{o} - \frac{A}{2}$ ?
(ii) Is ABPC a cyclic quadrilateral ?