Question

In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.

Answer 1

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°.