Question : ΔPQR is right angled at Q such that PQ = ( x - y ), QR = x, and PR = ( x + y ). S is a point on QR such that QS = PQ. The ratio QS : SR for any values of x and y is:
Option 1: 3 : 1
Option 2: 2 : 1
Option 3: 1 : 2
Option 4: 1 : 3
Correct Answer: 3 : 1
Solution : Given:$PR=(x+y)$ $PQ=(x-y)$ $QR=x$ And $QS=PQ$ In $\triangle PQR$ Use Pythagoras' theorem: $PR^2=PQ^2+QR^2$ ⇒ $(x+y)^2=(x-y)^2+x^2$ ⇒ $x^2+y^2+2xy=x^2+y^2-2xy+x^2$ ⇒ $4xy=x^2$ ⇒ $x=4y$ Now, $PR=(x+y)=4y+y=5y$ $PQ=(x-y)=4y-y=3y$ $QR=x=4y$ Thus, the ratio of sides of PR : QR : PQ = 5 : 4 : 3 Let PR = 5k, QR = 4k, and PQ = 3k. Since, QS = PQ = 3k $\therefore$ $SR = QR – PQ$ = 4k – 3k = k So, $QS : SR = 3k : k = 3 : 1$ Hence, the correct answer is 3 : 1.
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