Try this problem from the Singapore Mathematics Olympiad, SMO, 2010 based on the application of the Pythagoras Theorem.
The figure below shows a circle with diameter AB. C ad D are points on the circle on the same side of AB such that BD bisects \(\angle {CBA}\). The chords AC and BD intersect at E. It is given that AE = 169 cm and EC = 119 cm. If ED = x cm, find the value of x.
Circle
Pythagoras Theorem
2D - Geometry
Answer: 65
Singapore Mathematical Olympiad
Challenges and Thrills - Pre - College Mathematics
If you get stuck in this problem this is the first hint we can start with :
As BE intersect \(\angle {CBA}\) we have \(\frac {BC}{BA} = \frac {EC}{EA} = \frac {119}{169}\)
Thus we can let BC = 119 y and BA = 169 y .
Since \(\angle {BCA} = 90 ^\circ\).
Then try to do the rest of the problem ......................................................
If we want to continue from the last hint we have :
Apply Pythagoras Theorem ,
\(AB ^2 = AC^2 + BC ^2\)
\((169y)^2 = (169 + 119)^2 + (119y)^2\)
\(y^2 (169-119)(169+119) = (169+119)^2\)
\(y^2 = \frac {169+119}{169-119} = \frac {144}{25}\)
\(y = \frac {12}{5}\)
In the last hint:
Hence , from triangle BCE , we have BE = \(\sqrt{119^2 + (119y)^2} = 119 \times \frac {13}{5}\)
Finally , note that \(\triangle {ADE}\) and \(\triangle {BCE}\) are similar , so we have
ED = \(\frac {AE \times CE}{BE} = \frac {169 \times 119}{119 \times \frac {13}{5}} = 65 \) cm .
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