Difference between revisions of "1967 AHSME Problems/Problem 32"
AlcumusGuy (talk | contribs) (Fixed problem statement.) |
Lucasxia01 (talk | contribs) (→Solution) |
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== Solution == | == Solution == | ||
− | <math>\ | + | We note that <math>BO \cdot DO = AO \cdot CO = 24</math>. This is the Power of a Point Theorem which only happens to chords in circles. Therefore, we conclude that <math>ABCD</math> is cyclic. We can proceed with similar triangles. Because of inscribed angles, <math>\triangle ABO \simeq \triangle DCO</math> and <math>\triangle ADO \simeq \triangle BCO</math>. We find <math>\frac{CD}{AB} = \frac{3}{4} \implies CD = \frac{9}{2}</math> with the first similarity and <math>\frac{BC}{AD} = \frac{3}{6} \implies BC = \frac{AD}{2}</math> with the second similarity. Now, we can apply Ptolemy's theorem which states that in a cyclic quadrilateral, <math>AB \cdot CD + AD \cdot BC = AC \cdot BD</math>. We can plug in out values to get <math>6 \cdot \frac{9}{2} + AD \cdot \frac{AD}{2} = 11 \cdot 10 = 110</math>. We solve for <math>AD</math> to get <math>AD = \boxed{\textbf{(E) } \sqrt{166}}</math>. |
+ | - [b] lucasxia01[/b] | ||
== See also == | == See also == |
Revision as of 14:13, 19 June 2016
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 | |
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 |
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