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Difference between revisions of "2024 AIME I Problems/Problem 10"

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==Problem==
 
==Problem==
 
Let <math>ABC</math> be a triangle inscribed in circle <math>\omega</math>. Let the tangents to <math>\omega</math> at <math>B</math> and <math>C</math> intersect at point <math>P</math>, and let <math>\overline{AP}</math> intersect <math>\omega</math> at <math>D</math>. Find <math>AD</math>, if <math>AB=5</math>, <math>BC=9</math>, and <math>AC=10</math>.
 
Let <math>ABC</math> be a triangle inscribed in circle <math>\omega</math>. Let the tangents to <math>\omega</math> at <math>B</math> and <math>C</math> intersect at point <math>P</math>, and let <math>\overline{AP}</math> intersect <math>\omega</math> at <math>D</math>. Find <math>AD</math>, if <math>AB=5</math>, <math>BC=9</math>, and <math>AC=10</math>.
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==Solution 1==
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From the tangency condition we have <math>\let\angle BCD = \let\angle CBD = \let\angle A</math>. With LoC we have <math>\cos(A) = \frac{25+100-81}{2*5*10} = \frac{11}{25}</math> and <math>\cos(B) = \frac{81+25-100}{2*9*5} = \frac{1}{15}</math>. Then, <math>CD = \frac{\frac{9}{2}}{\cos(A)} = \frac{225}{22}</math>. Using LoC we can find <math>AD</math>: <math>AD^2 = AC^2 + CD^2 - 2(AC)(CD)\cos(A+C) = 10^2+(\frac{225}{22})^2 + 2(10)\frac{225}{22}\cos(B) = 100 + \frac{225^2}{22^2} + 2(10)\frac{225}{22}*\frac{1}{15} = \frac{5^4*13^2}{484}</math>. Thus, <math>AD = \frac{5^2*13}{22}</math>. By Power of a Point, <math>DP*AD = CD^2</math> so <math>DP*\frac{5^2*13}{22} = (\frac{225}{22})^2</math> which gives <math>DP = \frac{5^2*9^2}{13*22}</math>. Finally, we have <math>AP = AD - DP = \frac{5^2*13}{22} - \frac{5^2*9^2}{13*22} = \frac{100}{13} \rightarrow \boxed{113}</math>.
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~angie.
  
 
==See also==
 
==See also==

Revision as of 19:34, 2 February 2024

Problem

Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $P$, and let $\overline{AP}$ intersect $\omega$ at $D$. Find $AD$, if $AB=5$, $BC=9$, and $AC=10$.

Solution 1

From the tangency condition we have $\let\angle BCD = \let\angle CBD = \let\angle A$. With LoC we have $\cos(A) = \frac{25+100-81}{2*5*10} = \frac{11}{25}$ and $\cos(B) = \frac{81+25-100}{2*9*5} = \frac{1}{15}$. Then, $CD = \frac{\frac{9}{2}}{\cos(A)} = \frac{225}{22}$. Using LoC we can find $AD$: $AD^2 = AC^2 + CD^2 - 2(AC)(CD)\cos(A+C) = 10^2+(\frac{225}{22})^2 + 2(10)\frac{225}{22}\cos(B) = 100 + \frac{225^2}{22^2} + 2(10)\frac{225}{22}*\frac{1}{15} = \frac{5^4*13^2}{484}$. Thus, $AD = \frac{5^2*13}{22}$. By Power of a Point, $DP*AD = CD^2$ so $DP*\frac{5^2*13}{22} = (\frac{225}{22})^2$ which gives $DP = \frac{5^2*9^2}{13*22}$. Finally, we have $AP = AD - DP = \frac{5^2*13}{22} - \frac{5^2*9^2}{13*22} = \frac{100}{13} \rightarrow \boxed{113}$.

~angie.

See also

2024 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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