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Difference between revisions of "2011 AMC 10B Problems/Problem 17"

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[[Category: Introductory Geometry Problems]]
 
[[Category: Introductory Geometry Problems]]
  
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== Solution ==
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== Solution 1==
  
 
We can let <math>\angle AEB</math> be <math>4x</math> and <math>\angle ABE</math> be <math>5x</math> because they are in the ratio <math>4 : 5</math>. When an [[inscribed angle]] contains the [[diameter]], the inscribed angle is a [[right angle]]. Therefore by triangle sum theorem, <math>4x+5x+90=180 \longrightarrow x=10</math> and <math>\angle ABE = 50</math>.
 
We can let <math>\angle AEB</math> be <math>4x</math> and <math>\angle ABE</math> be <math>5x</math> because they are in the ratio <math>4 : 5</math>. When an [[inscribed angle]] contains the [[diameter]], the inscribed angle is a [[right angle]]. Therefore by triangle sum theorem, <math>4x+5x+90=180 \longrightarrow x=10</math> and <math>\angle ABE = 50</math>.
  
 
<math>\angle ABE = \angle BED</math> because they are alternate interior angles and <math>\overline{AB} \parallel \overline{ED}</math>. Opposite angles in a [[cyclic]] quadrilateral are [[supplementary]], so <math>\angle BED + \angle BCD = 180</math>. Use substitution to get <math>\angle ABE + \angle BCD = 180 \longrightarrow 50 + \angle BCD = 180 \longrightarrow \angle BCD = \boxed{\textbf{(C)} 130}</math>
 
<math>\angle ABE = \angle BED</math> because they are alternate interior angles and <math>\overline{AB} \parallel \overline{ED}</math>. Opposite angles in a [[cyclic]] quadrilateral are [[supplementary]], so <math>\angle BED + \angle BCD = 180</math>. Use substitution to get <math>\angle ABE + \angle BCD = 180 \longrightarrow 50 + \angle BCD = 180 \longrightarrow \angle BCD = \boxed{\textbf{(C)} 130}</math>
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==Solution 2==
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Note <math>\angle ABE = \angle BED=50</math> as before. The sum of the interior angles for quadrilateral <math>EBCD</math> is <math>360</math>. Denote the center of the circle as <math>P</math>. <math>\angle PDE = \angle PED = 50</math>. Denote <math>\angle PDC = \angle PCD = x</math> and <math>\angle PBC = \angle PCB = y</math>. We wish to find <math>\angle BCD = x+y</math>. Our equation is <math>(\angle PDE +\angle PED)+(\angle PDC+\angle PCD)+(\angle PBC + \angle PCB) = 360 \longrightarrow 2(50) + 2x +2y = 360</math>. Our final equation becomes <math>2(x+y)+100 = 360</math>. After subtracting <math>100</math> and dividing by <math>2</math>, our answer becomes <math>x+y=\boxed{\textbf{(C)} 130}</math>
  
 
== See Also==
 
== See Also==

Revision as of 20:55, 24 December 2017

Problem

In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4 : 5$. What is the degree measure of angle $BCD$?

[asy] unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real r=3; pair A=(-3cos(80),-3sin(80)); pair D=(3cos(80),3sin(80)), C=(-3cos(80),3sin(80)); pair O=(0,0), E=(-3,0), B=(3,0); path outer=Circle(O,r); draw(outer); draw(E--B); draw(E--A); draw(B--A); draw(E--D); draw(C--D); draw(B--C); pair[] ps={A,B,C,D,E,O}; dot(ps); label("$A$",A,N); label("$B$",B,NE); label("$C$",C,S); label("$D$",D,S); label("$E$",E,NW); label("$$",O,N); [/asy]

$\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 125 \qquad\textbf{(C)}\ 130 \qquad\textbf{(D)}\ 135 \qquad\textbf{(E)}\ 140$

Solution 1

We can let $\angle AEB$ be $4x$ and $\angle ABE$ be $5x$ because they are in the ratio $4 : 5$. When an inscribed angle contains the diameter, the inscribed angle is a right angle. Therefore by triangle sum theorem, $4x+5x+90=180 \longrightarrow x=10$ and $\angle ABE = 50$.

$\angle ABE = \angle BED$ because they are alternate interior angles and $\overline{AB} \parallel \overline{ED}$. Opposite angles in a cyclic quadrilateral are supplementary, so $\angle BED + \angle BCD = 180$. Use substitution to get $\angle ABE + \angle BCD = 180 \longrightarrow 50 + \angle BCD = 180 \longrightarrow \angle BCD = \boxed{\textbf{(C)} 130}$

Solution 2

Note $\angle ABE = \angle BED=50$ as before. The sum of the interior angles for quadrilateral $EBCD$ is $360$. Denote the center of the circle as $P$. $\angle PDE = \angle PED = 50$. Denote $\angle PDC = \angle PCD = x$ and $\angle PBC = \angle PCB = y$. We wish to find $\angle BCD = x+y$. Our equation is $(\angle PDE +\angle PED)+(\angle PDC+\angle PCD)+(\angle PBC + \angle PCB) = 360 \longrightarrow 2(50) + 2x +2y = 360$. Our final equation becomes $2(x+y)+100 = 360$. After subtracting $100$ and dividing by $2$, our answer becomes $x+y=\boxed{\textbf{(C)} 130}$

See Also

2011 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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All AMC 10 Problems and Solutions

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