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

Revision as of 03:25, 10 March 2023

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 $\tan(\angle ABC)$ ?

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

Diagram

$[asy] /* Made by MRENTHUSIASM */ size(250); real xMin = -15; real xMax = 15; real yMin = -17; real yMax = 17; draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2)); real f(real x) { return x^2+2*x-15; } draw(graph(f,-6.75,4.75),red); pair A, B, C; A = (-5,0); B = (0,-15); C = (3,0); draw(A--B--C); dot("$A$",A,1.5NW,linewidth(4.5)); dot("$B$",B,1.5SE,linewidth(4.5)); dot("$C$",C,1.5NE,linewidth(4.5)); label("$y=x^2+2x-15$",(12,9),red); [/asy]$ ~MRENTHUSIASM

Solution 1 (Dot Product)

First, find $A=(-5,0)$ , $B=(0,-15)$ , and $C=(3,0)$ . Create vectors $\overrightarrow{BA}$ and $\overrightarrow{BC}.$ These can be reduced to $\langle -1, 3 \rangle$ and $\langle 1, 5 \rangle$ , respectively. Then, we can use the dot product to calculate the cosine of the angle (where $\theta=\angle ABC$ ) between them:

$\begin{align*} \langle -1, 3 \rangle \cdot \langle 1, 5 \rangle = 15-1 &= \sqrt{10}\sqrt{26}\cos(\theta),\\ \implies \cos (\theta) &= \frac{7}{\sqrt{65}}. \end{align*}$

Thus, $\tan(\angle ABC) = \sqrt{\frac{65}{49}-1}= \boxed{\textbf{(E)}\ \frac{4}{7}}.$

~Indiiiigo

Solution 2

Note that $y=x^2+2x-15$ 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 point $O=(0, 0)$ . It follows that $\triangle AOB$ and $\triangle BOC$ are right triangles.

We have $\tan(\angle ABC) = \tan(\angle ABO + \angle OBC) = \frac{\tan(\angle ABO) + \tan(\angle OBC)}{1 - \tan(\angle ABO) \cdot \tan(\angle OBC)} = \frac{\frac15 + \frac13}{1 - \frac1{15}} = \boxed{\textbf{(E)}\ \frac{4}{7}}.$ Alternatively, we can use the Pythagorean Theorem to find that $AB = 5 \sqrt{10}$ and $BC = 3 \sqrt{26}$ and then use the $A = \frac12 ab \sin \angle C$ area formula for a triangle and the Law of Cosines to find $\tan(\angle ABC)$ .

Solution 3

Like above, we set $A$ to $(-5,0)$ , $B$ to $(0, -15)$ , and $C$ to $(3,0)$ , then finding via the Pythagorean Theorem that $AB = 5 \sqrt{10}$ and $BC = 3 \sqrt{26}$ . Using the Law of Cosines, we see that $\cos(\angle ABC) = \frac{AB^2 + BC^2 - AC^2}{2 AB BC} = \frac{250 + 234 - 64}{15 \sqrt{260}} = \frac{7}{\sqrt{65}}.$ Then, we use the identity $\tan^2(x) = \sec^2(x) - 1$ to get $\tan(\angle ABC) = \sqrt{\frac{65}{49} - 1} = \boxed{\textbf{(E)}\ \frac{4}{7}}.$

~ jamesl123456

Solution 4

We can reflect the figure, but still have the same angle. This problem is the same as having points $D(0,0)$ , $E(3,15)$ , and $F(-5,15)$ , where we're solving for angle FED. We can use the formula for $\tan(a-b)$ to solve now where $a$ is the $x$ -axis to angle $F$ and $b$ is the $x$ -axis to angle $E$ . $\tan(a) = \textrm{slope of line }DF = -3$ and $\tan(B) = \textrm{slope of line }DE = 5$ . Plugging these values into the $\tan(a-b)$ formula, we get $(-3-5)/(1+(-3\cdot 5))$ which is $\boxed{\textbf{(E)}\ \frac{4}{7}}.$

~mathboy100 (minor LaTeX edits)

Solution 5

We use the formula $[ABC]=\frac{1}{2}ab\sin{C}.$

Note that $\triangle ABC$ has side-lengths $AB=5\sqrt{10}$ and $BC=3\sqrt{26}$ from Pythagorean theorem, with the area being $\frac12\cdot8\cdot15.$

We equate the areas together to get: $\frac12\cdot8\cdot15=\frac12\cdot5\sqrt{10}\cdot3\sqrt{26}\cdot\sin{B},$ from which $\sin{B}=\frac{8}{\sqrt{260}}.$

From Pythagorean Identity, $\cos{B}=\frac{14}{\sqrt{260}}.$

Then we use $\tan{B}=\frac{\sin{B}}{\cos{B}}$ , to obtain $\tan{B}=\frac{8}{14}=\boxed{\textbf{(E)}\ \frac{4}{7}}.$

- SAHANWIJETUNGA

Solution 6 (Complex numbers)

$[asy] /* Made by MRENTHUSIASM */ size(300); real xMin = -15; real xMax = 15; real yMin = -17; real yMax = 17; draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2)); real f(real x) { return x^2+2*x-15; } draw(graph(f,-6.75,4.75),red); pair A, B, C, O; A = (-5,0); B = (0,-15); C = (3,0); O = origin; markscalefactor=0.1; draw(rightanglemark(B,O,C)); draw(A--B--C); dot("$A$",A,1.5SW,linewidth(4.5)); dot("$B$",B,1.5SE,linewidth(4.5)); dot("$C$",C,1.5SE,linewidth(4.5)); dot("$O$",O,1.5SW,linewidth(4.5)); label("$y=x^2+2x-15$",(12,9),red); label("$5$",(-2.5,0),1.5N); label("$3$",(1.5,0),1.5N); label("$15$",(0,-6),W); label("$\theta$",(0,-15),9*dir(100)); label("$\phi$",(0,-15),9*dir(84)); [/asy]$ From $x^2 + 2x - 15 = (x-3)(x+5)$ , we may assume, without loss of generality, that $x$ -intercepts of the given parabola are $A(-5,0)$ and $C(3,0)$ . And, point $B$ has coordinates $(0,-15)$ . Consider complex numbers $z = 5 + i$ and $w = 3 + i$ whose arguments are $\theta \coloneqq \angle OBA$ and $\phi \coloneqq \angle OBC$ , respectively. Notice that $\angle ABC = \theta + \phi$ is the argument of the product $zw$ which is $zw = (5+i)(3+i) = 14 + 8i.$ Hence $\tan \angle ABC = \frac{\operatorname{Im}(zw)}{\operatorname{Re}(zw)} = \frac{8}{14} = \boxed{\textbf{(E)}\ \frac{4}{7}}.$

~VensL.

2022 AMC 12B (Problems • Answer Key • Resources)
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