1967 AHSME Problems/Problem 32
Problem
In quadrilateral with diagonals
and
, intersecting at
,
,
,
,
, and
. The length of
is:
Solution
We note that . This is the Power of a Point Theorem which only happens to chords in circles. Therefore, we conclude that
is cyclic. We can proceed with similar triangles. Because of inscribed angles,
and
. We find
with the first similarity and
with the second similarity. Now, we can apply Ptolemy's theorem which states that in a cyclic quadrilateral,
. We can plug in out values to get
. We solve for
to get
.
- [b] lucasxia01[/b]
See also
1967 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 31 |
Followed by Problem 33 | |
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