Difference between revisions of "2001 AMC 12 Problems/Problem 24"

Revision as of 10:42, 25 August 2022

Problem

In $\triangle ABC$ , $\angle ABC=45^\circ$ . Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$ . Find $\angle ACB.$

$\text{(A) }54^\circ \qquad \text{(B) }60^\circ \qquad \text{(C) }72^\circ \qquad \text{(D) }75^\circ \qquad \text{(E) }90^\circ$

Solution 1

$[asy] unitsize(2cm); defaultpen(0.8); pair A=(0,0), B=(4,0), D=intersectionpoint( A -- (dir(15)*100), B -- (B+100*dir(135)) ), C=B+3*(D-B); pair ortho=rotate(-50)*(D-A); pair E=extension(C, ortho, A, D); draw(A--B); draw(B--C); draw(A--C); draw(C--E); draw(E--D); draw(A--D); draw(B--E, dotted); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE); label("$E$",E,S); [/asy]$

We start with the observation that $\angle ADB = 180^\circ - 15^\circ - 45^\circ = 120^\circ$ , and $\angle ADC = 15^\circ + 45^\circ = 60^\circ$ .

We can draw the height $CE$ from $C$ onto $AD$ . In the triangle $CED$ , we have $\frac {ED}{CD} = \cos EDC = \cos 60^\circ = \frac 12$ . Hence $ED = CD/2$ .

By the definition of $D$ , we also have $BD=CD/2$ , therefore $BD=DE$ . This means that the triangle $BDE$ is isosceles, and as $\angle BDE=120^\circ$ , we must have $\angle BED = \angle EBD = 30^\circ$ .

Then we compute $\angle ABE = 45^\circ - 30^\circ = 15^\circ$ , thus $\angle ABE = \angle BAE$ and the triangle $ABE$ is isosceles as well. Hence $AE=BE$ .

Now we can note that $\angle DCE = 180^\circ - 90^\circ - 60^\circ = 30^\circ$ , hence also the triangle $EBC$ is isosceles and we have $BE=CE$ .

Combining the previous two observations we get that $AE=EC$ , and as $\angle AEC=90^\circ$ , this means that $\angle CAE = \angle ACE = 45^\circ$ .

Finally, we get $\angle ACB = \angle ACE + \angle ECD = 45^\circ + 30^\circ = 75^\circ \boxed{D}$ .

Solution 2

Draw a good diagram! Now, let's call $BD=t$ , so $DC=2t$ . Given the rather nice angles of $\angle ABD = 45^\circ$ and $\angle ADC = 60^\circ$ as you can see, let's do trig. Drop an altitude from $A$ to $BC$ ; call this point $H$ . We realize that there is no specific factor of $t$ we can call this just yet, so let $AH=kt$ . Notice that in $\triangle{ABH}$ we get $BH=kt$ . Using the 60-degree angle in $\triangle{ADH}$ , we obtain $DH=\frac{\sqrt{3}}{3}kt$ . The comparable ratio is that $BH-DH=t$ . If we involve our $k$ , we get:

$kt(\frac{3}{3}-\frac{\sqrt{3}}{3})=t$ . Eliminating $t$ and removing radicals from the denominator, we get $k=\frac{3+\sqrt{3}}{2}$ . From there, one can easily obtain $HC=3t-kt=\frac{3-\sqrt{3}}{2}t$ . Now we finally have a desired ratio. Since $\tan\angle ACH = 2+\sqrt{3}$ upon calculation, we know that $\angle ACH$ can be simplified. Indeed, if you know that $\tan(75)=2+\sqrt{3}$ or even take a minute or two to work out the sine and cosine using $\sin(x)^2+\cos(x)^2=1$ , and perhaps the half- or double-angle formulas, you get $\boxed{75^\circ}$ .

Solution 3

Without loss of generality, we can assume that $BD = 1$ and $CD = 2$ . As above, we are able to find that $\angle ADC = 60^\circ$ and $\angle ADB = 120^\circ$ .

Using Law of Sines on triangle $ADB$ , we find that $\frac{1}{\sin15^\circ} = \frac{AD}{\sin 45^\circ} = \frac{AB}{\sin 120^\circ}.$ Since we know that $\sin 15^\circ = \frac{\sqrt{6}-\sqrt{2}}{4},$ $\sin 45^\circ = \frac{\sqrt{2}}{2},$ $\sin 120^\circ = \frac{\sqrt{3}}{2},$ we can compute $AD$ to equal $1+\sqrt{3}$ and $AB$ to be $\frac{3\sqrt{2}+\sqrt{6}}{2}$ .

Next, we apply Law of Cosines to triangle $ADC$ to see that $AC^2 = (1+\sqrt{3})^2 + 2^2 - (2)(1+\sqrt{3})(2)(\cos 60^\circ).$ Simplifying the right side, we get $AC^2 = 6$ , so $AC = \sqrt{6}$ .

Now, we apply Law of Sines to triangle $ABC$ to see that $\frac{\sqrt{6}}{\sin 45^\circ} = \frac{\frac{3\sqrt{2}+\sqrt{6}}{2}}{\sin ACB}.$ After rearranging and noting that $\sin 45^\circ = \frac{\sqrt{2}}{2}$ , we get $\sin ACB = \frac{\sqrt{6}+3\sqrt{2}}{4\sqrt{3}}.$

Dividing the right side by $\sqrt{3}$ , we see that $\sin ACB = \frac{\sqrt{6}+\sqrt{2}}{4},$ so $\angle ACB$ is either $75^\circ$ or $105^\circ$ . Since $105^\circ$ is not a choice, we know $\angle ACB = \boxed{75^\circ}$ .

Note that we can also confirm that $\angle ACB \neq 105^\circ$ by computing $\angle CAB$ with Law of Sines.

Solution 4(FAST)

Note that $\angle{ADB} = 120$ and $\angle{ADC} = 60$ . Seeing these angles makes us think of 30-60-90 triangles. Let $E$ be the foot of the altitude from $A$ to $BC$ . This means $\angle{DAE} = 30$ and $\angle{BAE} = 45$ . Let $BD = x$ and $DE = y$ . This means $AE = y\sqrt{3}$ and since $AE = BE$ we know that $x + y = y\sqrt{3}$ . This means $y = \frac{x(\sqrt{3} + 1)}{2}$ . This gives $CE = \frac{4x - x(\sqrt{3} + 1)}{2}$ . Note that $\tan{\angle{ACE}} = \frac{AE}{CE} = 2 + \sqrt{3}$ . Looking that the answer options we see that $\tan{75^\circ} = 2 + \sqrt{3}$ . This means the answer is $D$ . ~coolmath_2018

Solution 5 (Law of Sines)

$\angle ADB = 120^\circ$ , $\angle ADC = 60^\circ$ , $\angle DAB = 15^\circ$ , let $\angle ACB = \theta$ , $\angle DAC = 120^\circ - \theta$

By the Law of Sines we have $\frac{CD}{\sin(120^\circ - \theta)} = \frac{AD}{\sin \theta}$ , $\frac{BD}{\sin15^\circ} = \frac{AD}{\sin45^\circ}$

$\frac{BD}{CD} \cdot \frac{\sin(120^\circ - \theta)}{\sin15^\circ} = \frac{\sin \theta}{\sin45^\circ}$

$\frac{1}{2} \cdot \frac{\sin(120^\circ - \theta)}{\sin15^\circ} = \frac{\sin \theta}{\sin45^\circ}$

By the Triple-angle identities, $\sin 45^\circ = 3\sin15^\circ -4\sin^3 15^\circ$

$\frac{\sin(120^\circ - \theta)}{2} = \frac{\sin \theta}{3\ -4\sin^2 15^\circ}$

$\sin^2 15^\circ = \frac{1 - \cos 30^\circ}{2} = \frac{1 - \frac{\sqrt{3}}{2}}{2} = \frac{2 - \sqrt{3}}{4}$

$\frac{\sin(120^\circ - \theta)}{2} = \frac{\sin \theta}{\frac{2 - \sqrt{3}}{4}} = \frac{\sin \theta}{1+\sqrt{3}}$

$\frac{\sin \theta}{\sin(120^\circ - \theta)} = \frac{1+\sqrt{3}}{2}$

$\sin(120^\circ - \theta) = \sin120^\circ \cos\theta - \sin\theta \cos120^\circ = \frac{\sqrt{3}}{2} \cdot \cos\theta + \frac{1}{2} \cdot \sin\theta$

$\frac{\sin \theta}{\sin(120^\circ - \theta)} = \frac{\sin \theta}{\frac{\sqrt{3}}{2} \cdot \cos\theta + \frac{\sqrt{1}}{2} \cdot \sin\theta} = \frac{1+\sqrt{3}}{2}$

$\frac{\sqrt{3} \cdot \cos\theta + \sin\theta}{2 \sin \theta} = \frac{2}{1+\sqrt{3}}$

$\frac{\sqrt{3}}{2} \cdot \cot \theta = \frac{2}{1+\sqrt{3}} - \frac{1}{2} = \frac{2 - \sqrt{3}}{2 + 2 \sqrt{3}}$

$\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$

Suppose $\cos \theta = k(\sqrt{3} - 1)$ , and $\sin \theta = k(\sqrt{3} + 1)$

$\sin^2 \theta + \cos^2 \theta = 1$ , $k^2(\sqrt{3} + 1)^2 + k^2(\sqrt{3} - 1)^2 = 8k^2 = 1$ , $k = \frac{1}{2 \sqrt{2}}$

$\sin \theta = \frac{\sqrt{3} + 1}{2 \sqrt{2}}$ , $\cos \theta = \frac{\sqrt{3} - 1}{2 \sqrt{2}}$

$\sin 2 \theta = 2 \cdot \frac{\sqrt{3} + 1}{2 \sqrt{2}} \cdot \frac{\sqrt{3} - 1}{2 \sqrt{2}} = \frac{1}{2}$

Two possible values of $2 \theta$ are $150^\circ$ and $30^\circ$ . However we can rule out $30^\circ$ because $\cos \15^\circ$ (Error compiling LaTeX. Unknown error_msg) is positive, while $\cos \theta$ is negative.

Therefore $2 \theta = 150^\circ$ , $\angle ACB = \boxed{\textbf{(D) } 75^\circ }$

~isabelchen

@@ Line 95: / Line 95: @@
 <math>\frac{\sin \theta}{\sin(120^\circ - \theta)} = \frac{1+\sqrt{3}}{2}</math>
-<math>\sin(120^\circ - \theta) = \sin120^\circ \cos\theta - \sin\theta \cos120^\circ = \frac{\sqrt{3}}{2} \cdot \cos\theta + \frac{\sqrt{1}}{2} \cdot \sin\theta</math>
+<math>\sin(120^\circ - \theta) = \sin120^\circ \cos\theta - \sin\theta \cos120^\circ = \frac{\sqrt{3}}{2} \cdot \cos\theta + \frac{1}{2} \cdot \sin\theta</math>
+<math>\frac{\sin \theta}{\sin(120^\circ - \theta)} = \frac{\sin \theta}{\frac{\sqrt{3}}{2} \cdot \cos\theta + \frac{\sqrt{1}}{2} \cdot \sin\theta} = \frac{1+\sqrt{3}}{2}</math>
+<math>\frac{\sqrt{3} \cdot \cos\theta + \sin\theta}{2 \sin \theta} = \frac{2}{1+\sqrt{3}}</math>
+<math>\frac{\sqrt{3}}{2} \cdot \cot \theta = \frac{2}{1+\sqrt{3}} - \frac{1}{2} = \frac{2 - \sqrt{3}}{2 + 2 \sqrt{3}}</math>
+<math>\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}</math>
+Suppose <math>\cos \theta = k(\sqrt{3} - 1)</math>, and <math>\sin \theta = k(\sqrt{3} + 1)</math>
+<math>\sin^2 \theta + \cos^2 \theta = 1</math>, <math>k^2(\sqrt{3} + 1)^2 + k^2(\sqrt{3} - 1)^2 = 8k^2 = 1</math>, <math>k = \frac{1}{2 \sqrt{2}}</math>
+<math>\sin \theta = \frac{\sqrt{3} + 1}{2 \sqrt{2}}</math>, <math>\cos \theta = \frac{\sqrt{3} - 1}{2 \sqrt{2}}</math>
+<math>\sin 2 \theta = 2 \cdot \frac{\sqrt{3} + 1}{2 \sqrt{2}} \cdot \frac{\sqrt{3} - 1}{2 \sqrt{2}} = \frac{1}{2}</math>
+Two possible values of <math>2 \theta</math> are <math>150^\circ</math> and <math>30^\circ</math>. However we can rule out <math>30^\circ</math> because <math>\cos \15^\circ</math> is positive, while <math>\cos \theta</math> is negative.
+Therefore <math>2 \theta = 150^\circ</math>, <math>\angle ACB = \boxed{\textbf{(D) } 75^\circ }</math>
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 == See Also ==

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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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