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Difference between revisions of "1956 AHSME Problems/Problem 49"

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<math>\angle AOB &= 180^\circ - \angle BAO - \angle ABO \\
 
<math>\angle AOB &= 180^\circ - \angle BAO - \angle ABO \\
−
&= 180^\circ - \frac{\angle BAT}{2} - \frac{\angle ABR}{2} \\
+
180^\circ - \frac{\angle BAT}{2} - \frac{\angle ABR}{2} \\
−
&= 180^\circ - \frac{180^\circ - \angle BAP}{2} - \frac{180^\circ - \angle ABP}{2} \\
+
180^\circ - \frac{180^\circ - \angle BAP}{2} - \frac{180^\circ - \angle ABP}{2} \\
−
&= \frac{\angle BAP + \angle ABP}{2}.</math>
+
\frac{\angle BAP + \angle ABP}{2}.</math>
  
 
Finally, from triangle <math>ABP</math>, <math>\angle BAP + \angle ABP = 180^\circ - \angle APB = 180^\circ - 40^\circ = 140^\circ</math>, so <cmath>\angle AOB = \frac{\angle BAP + \angle ABP}{2} = \frac{140^\circ}{2} = \boxed{70^\circ}.</cmath>
 
Finally, from triangle <math>ABP</math>, <math>\angle BAP + \angle ABP = 180^\circ - \angle APB = 180^\circ - 40^\circ = 140^\circ</math>, so <cmath>\angle AOB = \frac{\angle BAP + \angle ABP}{2} = \frac{140^\circ}{2} = \boxed{70^\circ}.</cmath>

Revision as of 12:38, 1 August 2020

Solution 1

First, from triangle $ABO$, $\angle AOB = 180^\circ - \angle BAO - \angle ABO$. Note that $AO$ bisects $\angle BAT$ (to see this, draw radii from $O$ to $AB$ and $AT,$ creating two congruent right triangles), so $\angle BAO = \angle BAT/2$. Similarly, $\angle ABO = \angle ABR/2$.

Also, $\angle BAT = 180^\circ - \angle BAP$, and $\angle ABR = 180^\circ - \angle ABP$. Hence,

$\angle AOB &= 180^\circ - \angle BAO - \angle ABO \\ 

180^\circ - \frac{\angle BAT}{2} - \frac{\angle ABR}{2} \\
180^\circ - \frac{180^\circ - \angle BAP}{2} - \frac{180^\circ - \angle ABP}{2} \\
\frac{\angle BAP + \angle ABP}{2}.$ (Error compiling LaTeX. Unknown error_msg)

Finally, from triangle $ABP$, $\angle BAP + \angle ABP = 180^\circ - \angle APB = 180^\circ - 40^\circ = 140^\circ$, so \[\angle AOB = \frac{\angle BAP + \angle ABP}{2} = \frac{140^\circ}{2} = \boxed{70^\circ}.\]

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