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Question
In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.
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Solution
PQ = `"RS"/(3)` = 8cm
⇒ PQ = 8cm and RS = 3 x 8 = 24cm
3ST = 4QT = 48cm
⇒ ST = `(48)/(3) = 16"cm" and "QT" = (48)/(4)` = 12cm
In ΔPTQ,
PT2 = PQ2 + QT2
= 82 + 122
= 64 + 144
= 208
In ΔRTS,
RT2 = RS2 + ST2
= 242 + 162
= 576 + 256
= 832
Now, PT2 + RT2
= 208 + 832
= 1040 .....(i)
Draw PU ⊥ RS and Join PR.
PU = SQ
= ST + TQ
= 16 + 12
= 28cm
RU = RS - US
= RS - PQ
= 24 - 8
= 16cm
In ΔRUP,
PR2 = RU2 + PU2
= 162 + 282
= 256 + 784
= 1040 ....(ii)
From (i) and (ii), we get
PT2 + RT2 = PR2
Thus, ∠RTP = 90°.
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