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Question Number 6750 by Tawakalitu. last updated on 22/Jul/16

	Answered by Yozzii last updated on 22/Jul/16

	$L e t ∠ Q P R = ∠ R P S = α > 0, ∠ P Q R = β > 0 a n d ∠ P S R = δ > 0 .$ $B y t h e s i n e r u l e, i n △ Q P R,$ $\frac{Q R}{s i n α} = \frac{Q P}{s i n ∠ P R Q} = \frac{3 a}{s i n (π - (α + β))}$ $\Rightarrow Q R = \frac{3 a s i n α}{s i n (α + β)} .$ $I n △ P R S, b y t h e s i n e r u l e w e h a v e$ $\frac{R S}{s i n α} = \frac{P S}{s i n ∠ P R S} = \frac{a}{s i n (π - (α + δ))}$ $R S = \frac{a s i n α}{s i n (α + δ)} .$ $∴ \frac{Q R}{R S} = \frac{3 a s i n α / s i n (α + β)}{a s i n α / s i n (α + δ)}$ $\frac{Q R}{R S} = \frac{3 s i n (α + δ)}{s i n (α + β)}$ $N o w, i n △ Q P S, ∠ Q P R + ∠ R P S + ∠ P S R + ∠ R Q P = π .$ $∴ α + α + β + δ = π \Rightarrow α + δ = π - (α + β) .$ $\Rightarrow \frac{Q R}{R S} = \frac{3 s i n (π - (α + β))}{s i n (α + β)}$ $\frac{Q R}{R S} = \frac{3 s i n (α + β)}{s i n (α + β)} = \frac{3}{1}$ $\Rightarrow Q R : R S = 3 : 1 \Rightarrow S R : (Q R + R S) = S R : S Q = 1 : (3 + 1) = 1 : 4$ $T h e p o i n t R d i v i d e s S Q i n t o t w o l e n g t h s s u c h t h a t$ $t h e l e n g t h Q R i s 3 t i m e s t h e l e n g t h R S,$ $w h i c h i m p l i e s t h a t R S i s o n e q u a r t e r$ $t h e e n t i r e s i d e S Q o f △ Q P S .$
	Commented by Tawakalitu. last updated on 22/Jul/16

	$T h a n k s s o m u c h . . i a p p r e c i a t e$
	Answered by sandy_suhendra last updated on 22/Jul/16

	$L e t ∠ Q P R = ∠ R P S = α$ $∠ Q R P = β s o ∠ S R P = (180 - β)$ $I n △ Q P R \Rightarrow \frac{Q R}{s i n α} = \frac{P Q}{s i n β} [s i n r u l e]$ $\frac{Q R}{s i n α} = \frac{3 a}{s i n β}$ $Q R = \frac{3 a s i n α}{s i n β}$ $I n △ P R S \Rightarrow \frac{R S}{s i n α} = \frac{P S}{s i n (180 - β)} \Rightarrow s i n (180 - β) = s i n β$ $\frac{R S}{s i n α} = \frac{a}{s i n β}$ $R S = \frac{a s i n α}{s i n β}$ $∴ Q R : R S = \frac{3 a s i n α}{s i n β} : \frac{a s i n α}{s i n β} = 3 : 1$ $S R : S Q = 1 : (1 + 3) = 1 : 4$ $(I h o p e t h i s a n s w e r i s m o r e s i m p l e)$
	Commented by Tawakalitu. last updated on 23/Jul/16

	$W o w t h a n k s$
	Answered by Rasheed Soomro last updated on 22/Jul/16

	Commented by Rasheed Soomro last updated on 23/Jul/16

	$A n s w e r u s i n g o n l y g e o m e t r y$ $i - e w i t h o u t u s i n g t r i g o n o m e r i c r a t i o s .$ $P a r t - I P r o v i n g P Q : P S = Q R : R S$ $\underline{G i v e n :}$ $△ P Q S, ∠ 1 = ∠ 2$ $\underline{C o n s t r u c t i o n :}$ $D r a w S T p a r a l l e l t o P R . L e t S T m e e t s p r o d u c e d$ $Q P a t T .$ $\underline{P r o o f :}$ $∠ 3 = ∠ 1 [C o r r e s p o n d i n g a n g l e s]$ $B u t ∠ 1 = ∠ 2 [G i v e n]$ $∴ ∠ 3 = ∠ 2 [T r a n s i t i v e p r o p e r t y o f e q u a l i t y]$ $B u t ∠ 2 = ∠ 4 [A l t e r n a t i v e a n g l e s]$ $∴ ∠ 3 = ∠ 4 [T r a n s i t i v e p r o p e r t y o f e q u a l i t y]$ $∴ P S = P T [O p p o s i t e s i d e s o f e q u a l a n g l e s]$ $N o w i n △ Q T S$ $∵ P R ∥ T S [C o n s t r u c t i o n]$ $∴ Q P : P T = Q R : R S [A t h e o r m]$ $B u t s i n c e P T = P S [P r o v e d a l r e a d y]$ $∴ Q P : P S = Q R : R S$ $P a r t - I I$ $\frac{Q P}{P S} = \frac{Q R}{R S} [P r o v e d]$ $\frac{Q P}{P S} + 1 = \frac{Q R}{R S} + 1 [A d d i n g 1 t o b o t h s i d e s]$ $O r \frac{Q P + P S}{P S} = \frac{Q R + R S}{R S}$ $\frac{Q P + P S}{P S} = \frac{Q S}{R S} [Q R + R S = Q S]$ $N o w Q P = 3 a a n d P S = a [G i v e n]$ $\frac{3 a + a}{a} = \frac{Q S}{R S}$ $O r \frac{Q S}{R S} = \frac{4}{1}$ $R S : Q S = 1 : 4$
	Commented by Tawakalitu. last updated on 23/Jul/16

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