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ICSE Class 9 Answered

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Asked by aryan2mumbai | 21 Oct, 2020, 08:40: AM
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Q: In the given figure below, 2 PQ = QS, R is the midpoint of QS, T is midpoint of PV and RV || SW. Prove that:
     (i) QT || SW
    (ii) PW = 3VW
 
Solution:
(i)
Since R is midpoint of QS, therefore QR = RS ... (1)
2PQ = QS
i.e. 2PQ = QR + RS = QR + QR ... From (1)
i.e. 2PQ = 2QR 
Therefore, PQ = QR ... (2)
Since, T is the midpoint of PV, therefore PT = TV ... (3)
Now, in triangle PRV, line segment QT divides the sides PR and PV in the same ratio which is 1:1
Using Basic proportionality theorem, we get
QT || RV
Now, as it is given that RV || SW, so we have
QT || SW
 
(ii)
From (2), PQ = QR = RS
Therefore, PR/RS = 2RS/RS = 2/1
In triangle PSW, RV || SW
So by using the converse of Basic proportionality theorem, we have
Line segment RV divides the sides PS and PW in the same ratio
Therefore, PV/VW = PR/RS = 2/1
PV = 2VW
Now, PW = PV + VW = 2VW + VW = 3VW
Answered by Renu Varma | 26 Oct, 2020, 11:10: AM

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