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Difference between revisions of "2022 AMC 12B Problems/Problem 14"

(Created page with "== Problem == The graph of <math>y=x^2+2x-15</math> intersects the <math>x</math>-axis at points <math>A</math> and <math>C</math> and the <math>y</math>-axis at point <math>...")
 
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== Solution ==
 
== Solution ==
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<math>y=x^2+2x-15</math> intersects the <math>x</math>-axis at points <math>(-5, 0)</math> and <math>(3, 0)</math>. Without loss of generality, let these points be <math>A</math> and <math>C</math> respectively. Also, the graph intersects the y-axis at point <math>B = (0, -15)</math>.
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Let the origin be point <math>O = (0, 0)</math>. Note that tiangles <math>AOB</math> and <math>COB</math> are right.
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<cmath>\tan(\angle ABC) = \frac{\sin (\angle ABC)}{\cos (\angle ABC)} = \frac{\sin(\angle ABO + \angle OBC)}{\cos(\angle ABO + \angle OBC)}</cmath>
  
 
== See Also ==
 
== See Also ==
 
{{AMC12 box|year=2022|ab=B|num-b=13|num-a=15}}
 
{{AMC12 box|year=2022|ab=B|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:58, 17 November 2022

Problem

The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is $\tan(\angle ABC)$?

$\textbf{(A)}\ \frac{1}{7} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{3}{7} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{4}{7} \qquad$

Solution

$y=x^2+2x-15$ intersects the $x$-axis at points $(-5, 0)$ and $(3, 0)$. Without loss of generality, let these points be $A$ and $C$ respectively. Also, the graph intersects the y-axis at point $B = (0, -15)$.

Let the origin be point $O = (0, 0)$. Note that tiangles $AOB$ and $COB$ are right.

\[\tan(\angle ABC) = \frac{\sin (\angle ABC)}{\cos (\angle ABC)} = \frac{\sin(\angle ABO + \angle OBC)}{\cos(\angle ABO + \angle OBC)}\]

See Also

2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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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