Difference between revisions of "2013 Mock AIME I Problems/Problem 5"
Thundercloak (talk | contribs) (→Solution 2) |
Thundercloak (talk | contribs) (→Solution 2) |
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Let <math>\angle DMC = \theta</math>. | Let <math>\angle DMC = \theta</math>. | ||
In triangle <math>DMC</math>, by the cosine rule, | In triangle <math>DMC</math>, by the cosine rule, | ||
− | <cmath>\cos \theta = \frac{3^2 + 4^2 - x^2}{24} = \frac{25 - x^2}{24}.</cmath> | + | <cmath>\cos \theta = \frac{3^2 + 4^2 - x^2}{24} = \frac{25 - x^2}{24}.</cmath> |
In triangle <math>CMB</math>, by the cosine rule, | In triangle <math>CMB</math>, by the cosine rule, | ||
− | <cmath>\cos (180^\circ - \theta) = \frac{4^2 + 8^2 - (2x)^2}{64} = \frac{80 - 4x^2}{64}.</cmath> | + | <cmath>\cos (180^\circ - \theta) = \frac{4^2 + 8^2 - (2x)^2}{64} = \frac{80 - 4x^2}{64}.</cmath> |
Since <math>\cos (180^\circ - \theta) = -\cos \theta</math>, we have: | Since <math>\cos (180^\circ - \theta) = -\cos \theta</math>, we have: | ||
<cmath>\frac{25 - x^2}{24} = -\frac{80 - 4x^2}{64}.</cmath> | <cmath>\frac{25 - x^2}{24} = -\frac{80 - 4x^2}{64}.</cmath> | ||
Solving for <math>x</math>, we get <math>x = \sqrt{22}</math>. | Solving for <math>x</math>, we get <math>x = \sqrt{22}</math>. | ||
− | Now, using the cosine rule in triangles <math>AMB</math> and <math>AMD</math> with <math>\cos \theta = \frac{1}{8}</math> (substituting <math>x = \sqrt{22}</math> | + | Now, using the cosine rule in triangles <math>AMB</math> and <math>AMD</math> with <math>\cos \theta = \frac{1}{8}</math> (substituting <math>x = \sqrt{22}</math>), we can find <math>AB</math> and <math>BC</math>. |
After calculations, we get <math>AB = 2x</math> and <math>BC = \frac{3x}{2}</math>. | After calculations, we get <math>AB = 2x</math> and <math>BC = \frac{3x}{2}</math>. | ||
The perimeter of <math>ABCD</math> is given by: | The perimeter of <math>ABCD</math> is given by: | ||
Line 79: | Line 79: | ||
Substituting <math>x = \sqrt{22}</math>, we get: | Substituting <math>x = \sqrt{22}</math>, we get: | ||
<cmath>\text{Perimeter} = \frac{13\sqrt{22}}{2}.</cmath> | <cmath>\text{Perimeter} = \frac{13\sqrt{22}}{2}.</cmath> | ||
− | Therefore, <math>p = 13, q = 22, r = 2</math>, and | + | Therefore, <math>p = 13, q = 22, r = 2</math>, and $p + q + r = 13 + 22 + 2 = 37. |
+ | Solution by Thunder Cloak | ||
== See also == | == See also == |
Latest revision as of 05:04, 25 August 2024
Contents
Problem
In quadrilateral , . Also, , and . The perimeter of can be expressed in the form where and are relatively prime, and is not divisible by the square of any prime number. Find .
Solution
Let , as in the diagram. Thus, from the problem, . Because , by Power of a Point, we know that is cyclic. Thus, we know that , so, by the congruency of vertical angles and subsequently AA Similarity, we know that . Thus, we have the proportion , or, by substitution, . Solving this equation for yields . Similarly, we know that , so, like before, we can see that . Thus, we have the proportion , or, by substitution, . Solving for yields .
Now, we can use Ptolemy's Theorem on cyclic and solve for : \begin{align*} x \cdot 2x + 2x \cdot \frac3 2 x &= (6+4)(8+3) \\ 5x^2 &= 110 \\ x^2 &= 22 \\ x &= \pm \sqrt{22} \end{align*} Because , . Thus, the perimeter of is . Thus, .
Solution 2
Consider the figure and notations from Solution 1.
Let . In triangle , by the cosine rule, In triangle , by the cosine rule, Since , we have: Solving for , we get . Now, using the cosine rule in triangles and with (substituting ), we can find and . After calculations, we get and . The perimeter of is given by: Substituting , we get: Therefore, , and $p + q + r = 13 + 22 + 2 = 37. Solution by Thunder Cloak