Triangles And Properties Of Triangle Question 34
Question: Each side of an equilateral triangle subtends an angle of $ 60{}^\circ $ at the top of a tower h m high located at the centre of the triangle. If a is the length of each of side of the triangle, then
Options:
A) $ 3a^{2}=2h^{2} $
B) $ 2a^{2}=3h^{2} $
C) $ a^{2}=3h^{2} $
D) $ 3a^{2}=h^{2} $
Show Answer
Answer:
Correct Answer: B
Solution:
[b] Let QT be the tower of height (h) I $ \Delta PRS. $ now, each triangle QPR, QRS, QSP ar equilateral. Thus QP=QS=QR=a. In $ \Delta QTP, $ $ QP^{2}=QT^{2}+PT^{2} $
$ \Rightarrow a^{2}=h^{2}+{{( \frac{a}{2}\sec 30{}^\circ )}^{2}} $
$ \Rightarrow a^{2}=h^{2}+\frac{a^{2}}{4}.\frac{4}{3} $
$ \Rightarrow a^{2}=h^{2}+\frac{a^{2}}{3} $
$ \Rightarrow a^{2}-\frac{a^{2}}{3}=h^{2} $
$ \Rightarrow \frac{3a^{2}-a^{2}}{3}=h^{2}\therefore 2a^{2}=3h^{2} $
BETA
