Difference between revisions of "2011 AMC 12B Problems/Problem 6"

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==Solution 3==
 
==Solution 3==
  
Let the center of the circle be <math>O</math>. The radii extending from <math>O</math> to the points of tangency <math>B</math> and <math>C</math> are both perpendicular to lines <math>AB</math> and <math>AC</math>. Thus <math>\angle ABO = \angle ACO = 90^{\circ}</math>, and quadrilateral <math>ABOC</math> is cyclic, since the opposite angles add to <math>180^{\circ}</math>. Since the ratio of the arc lengths is <math>2 : 3</math>, the shorter arc (the arc cut off by <math>\angle BOC</math>) is <math>144^{\circ}</math>. <math>\angle BOC</math> and <math>\angle BAC</math> must add up to <math>180^{\circ}</math>, so <math>\angle BAC = 180 - 144 = \boxed{36 \textbf{(C)}}</math>.
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Let the center of the circle be <math>O</math>. The radii extending from <math>O</math> to the points of tangency <math>B</math> and <math>C</math> are both perpendicular to lines <math>AB</math> and <math>AC</math>. Thus <math>\angle ABO = \angle ACO = 90^{\circ}</math>, and quadrilateral <math>ABOC</math> is cyclic, since the opposite angles add to <math>180^{\circ}</math>.  
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Since the ratio of the arc lengths is <math>2 : 3</math>, the shorter arc (the arc cut off by <math>\angle BOC</math>) is <math>144^{\circ}</math>.  
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<math>\angle BOC</math> and <math>\angle BAC</math> must add up to <math>180^{\circ}</math> since quadrilateral <math>ABOC</math> is cyclic, so <math>\angle BAC = 180 - 144 = \boxed{\textbf{(C) } 36}</math>.
  
 
-Solution by Joeya
 
-Solution by Joeya

Latest revision as of 12:35, 26 January 2021

Problem

Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\angle{BAC}$?

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60$

Solution

In order to solve this problem, use of the tangent-tangent intersection theorem (Angle of intersection between two tangents dividing a circle into arc length A and arc length B = 1/2 (Arc A° - Arc B°).

In order to utilize this theorem, the degree measures of the arcs must be found. First, set A (Arc length A) equal to 3d, and B (Arc length B) equal to 2d.

Setting 3d+2d = 360° will find d = 72°, and so therefore Arc length A in degrees will equal 216° and arc length B will equal 144°.

Finally, simply plug the two arc lengths into the tangent-tangent intersection theorem, and the answer:

1/2 (216°-144°) = 1/2 (72°) $=\boxed{36 \textbf{(C)}}.$

Solution 2

Let arc A be 3x and arc B be 2x. Then the angle formed by the tangents is $\frac{3x-2x}2 = \frac {x} 2$ by the arc length formula. Also note that $3x + 2x = 360$, which simplifies to $x= 72.$ Hence the angle formed by the tangents is equal to $\boxed{36 \textbf{(C)}}$.

Solution 3

Let the center of the circle be $O$. The radii extending from $O$ to the points of tangency $B$ and $C$ are both perpendicular to lines $AB$ and $AC$. Thus $\angle ABO = \angle ACO = 90^{\circ}$, and quadrilateral $ABOC$ is cyclic, since the opposite angles add to $180^{\circ}$.

Since the ratio of the arc lengths is $2 : 3$, the shorter arc (the arc cut off by $\angle BOC$) is $144^{\circ}$.

$\angle BOC$ and $\angle BAC$ must add up to $180^{\circ}$ since quadrilateral $ABOC$ is cyclic, so $\angle BAC = 180 - 144 = \boxed{\textbf{(C) } 36}$.

-Solution by Joeya

See also

2011 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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
All AMC 12 Problems and Solutions

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