∠ P S Q = 4 5 ∘ , ∠ P R Q = 3 0 ∘ angle mathrm{PSQ}=45^{circ}, angle mathrm{PRQ}=30^{circ} ∠ PSQ = 4 5 ∘ , ∠ PRQ = 3 0 ∘
From ΔPQS
tan 4 5 ∘ = P Q Q S an 45^{circ}=frac{mathrm{PQ}}{mathrm{QS}} tan 4 5 ∘ = QS PQ
⇒ 1 = x Q S ⇒ Q S = x Rightarrow 1=frac{x}{mathrm{QS}} Rightarrow mathrm{QS}=x ⇒ 1 = QS x ⇒ QS = x
From ΔPQR,
tan 3 0 ∘ = P Q Q R an 30^{circ}=frac{mathrm{PQ}}{mathrm{QR}} tan 3 0 ∘ = QR PQ
⇒ 1 3 = x x + y Rightarrow frac{1}{sqrt{3}}=frac{x}{x+y} ⇒ 3 1 = x + y x
⇒ 3 x = x + y Rightarrow sqrt{3} x=x+y ⇒ 3 x = x + y
⇒ 3 x − x = y Rightarrow sqrt{3} x-x=y ⇒ 3 x − x = y
⇒ x ( 3 − 1 ) = y Rightarrow x(sqrt{3}-1)=y ⇒ x ( 3 − 1 ) = y
⇒ x y = 1 3 − 1 = 3 + 1 ( 3 − 1 ) ( 3 + 1 ) Rightarrow frac{x}{y}=frac{1}{sqrt{3}-1}=frac{sqrt{3}+1}{(sqrt{3}-1)(sqrt{3}+1)} ⇒ y x = 3 − 1 1 = ( 3 − 1 ) ( 3 + 1 ) 3 + 1
= 3 + 1 3 − 1 = 3 + 1 2 =frac{sqrt{3}+1}{3-1}=frac{sqrt{3}+1}{2} = 3 − 1 3 + 1 = 2 3 + 1
