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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