Difference between revisions of "1967 AHSME Problems/Problem 32"
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== Problem == | == Problem == | ||
− | In | + | In rectangle <math>ABCD</math> with diagonals <math>AC</math> and <math>BD</math>, intersecting at <math>O</math>, <math>BO=4</math>, <math>OD = 6</math>, <math>AO=8</math>, <math>OC=3</math>, and <math>AB=6</math>. The length of <math>AD</math> is: |
<math>\textbf{(A)}\ 9\qquad | <math>\textbf{(A)}\ 9\qquad |
Revision as of 02:34, 4 January 2020
Problem
In rectangle 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
.
See also
1967 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 31 |
Followed by Problem 33 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.