Difference between revisions of "2021 AMC 12A Problems/Problem 24"
MRENTHUSIASM (talk | contribs) m (→Solution 3 (Weighted Averages and Similar Triangles): Fixed diagram.) |
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Semicircle <math>\Gamma</math> has diameter <math>\overline{AB}</math> of length <math>14</math>. Circle <math>\Omega</math> lies tangent to <math>\overline{AB}</math> at a point <math>P</math> and intersects <math>\Gamma</math> at points <math>Q</math> and <math>R</math>. If <math>QR=3\sqrt3</math> and <math>\angle QPR=60^\circ</math>, then the area of <math>\triangle PQR</math> equals <math>\tfrac{a\sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer not divisible by the square of any prime. What is <math>a+b+c</math>? | Semicircle <math>\Gamma</math> has diameter <math>\overline{AB}</math> of length <math>14</math>. Circle <math>\Omega</math> lies tangent to <math>\overline{AB}</math> at a point <math>P</math> and intersects <math>\Gamma</math> at points <math>Q</math> and <math>R</math>. If <math>QR=3\sqrt3</math> and <math>\angle QPR=60^\circ</math>, then the area of <math>\triangle PQR</math> equals <math>\tfrac{a\sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer not divisible by the square of any prime. What is <math>a+b+c</math>? | ||
− | <math>\textbf{(A) } 110\qquad\textbf{(B) } 114\qquad\textbf{(C) } 118\qquad\textbf{(D) } 122\qquad\textbf{(E) } 126 | + | <math>\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126</math> |
==Diagram== | ==Diagram== | ||
− | [[ | + | <asy> |
− | ~MRENTHUSIASM | + | /* Made by MRENTHUSIASM */ |
+ | size(300); | ||
+ | pair O, X, A, B, P, Q, R; | ||
+ | O = (0,0); | ||
+ | X = (4,3); | ||
+ | A = (-7,0); | ||
+ | B = (7,0); | ||
+ | P = (4,0); | ||
+ | Q = intersectionpoints(Circle(O,7),Circle(X,3))[0]; | ||
+ | R = intersectionpoints(Circle(O,7),Circle(X,3))[1]; | ||
+ | filldraw(P--Q--R--cycle,yellow); | ||
+ | dot("$\Gamma$",O,S); | ||
+ | dot("$\Omega$",X,E); | ||
+ | dot("$A$",A,SW); | ||
+ | dot("$B$",B,SE); | ||
+ | dot("$P$",P,S); | ||
+ | dot("$Q$",Q,E); | ||
+ | dot("$R$",R,N); | ||
+ | draw(arc(O, 7, 0, 180)^^A--B^^Circle(X,3)); | ||
+ | </asy> | ||
+ | ~MRENTHUSIASM | ||
==Solution 1 (Possible Without Trigonometry)== | ==Solution 1 (Possible Without Trigonometry)== | ||
Line 14: | Line 34: | ||
Alternatively, by the Inscribed Angle Theorem, <math>\triangle QRX</math> is a <math>30^\circ\text{-}30^\circ\text{-}120^\circ</math> triangle with base <math>QR=3\sqrt3.</math> Dividing <math>\triangle QRX</math> into two congruent <math>30^\circ\text{-}60^\circ\text{-}90^\circ</math> triangles, we get that the radius of <math>\odot X</math> is <math>XQ=XR=3</math> by the side-length ratios. | Alternatively, by the Inscribed Angle Theorem, <math>\triangle QRX</math> is a <math>30^\circ\text{-}30^\circ\text{-}120^\circ</math> triangle with base <math>QR=3\sqrt3.</math> Dividing <math>\triangle QRX</math> into two congruent <math>30^\circ\text{-}60^\circ\text{-}90^\circ</math> triangles, we get that the radius of <math>\odot X</math> is <math>XQ=XR=3</math> by the side-length ratios. | ||
− | Let <math>M</math> be the midpoint of <math>\overline{QR}.</math> By the Perpendicular Chord Bisector Converse, we have <math>\overline{ | + | Let <math>M</math> be the midpoint of <math>\overline{QR}.</math> By the Perpendicular Chord Bisector Converse, we have <math>\overline{OM}\perp\overline{QR}</math> and <math>\overline{XM}\perp\overline{QR}.</math> Together, points <math>O, X,</math> and <math>M</math> must be collinear. |
+ | |||
+ | By the SAS Congruence, we have <math>\triangle QXM\cong\triangle RXM,</math> both of which are <math>30^\circ\text{-}60^\circ\text{-}90^\circ</math> triangles. By the side-length ratios, we obtain <math>RM=\frac{3\sqrt3}{2}, RX=3,</math> and <math>XM=\frac{3}{2}.</math> By the Pythagorean Theorem on right <math>\triangle ORM,</math> we get <math>OM=\frac{13}{2}</math> and <math>OX=OM-XM=5.</math> By the Pythagorean Theorem on right <math>\triangle OXP,</math> we get <math>OP=4.</math> | ||
− | + | Let <math>C</math> be the foot of the perpendicular from <math>P</math> to <math>\overline{QR},</math> and <math>D</math> be the foot of the perpendicular from <math>X</math> to <math>\overline{PC},</math> as shown below: | |
− | [[ | + | <asy> |
− | + | /* Made by MRENTHUSIASM */ | |
+ | size(300); | ||
+ | pair O, X, A, B, P, Q, R, M, C, D; | ||
+ | O = (0,0); | ||
+ | X = (4,3); | ||
+ | A = (-7,0); | ||
+ | B = (7,0); | ||
+ | P = (4,0); | ||
+ | Q = intersectionpoints(Circle(O,7),Circle(X,3))[0]; | ||
+ | R = intersectionpoints(Circle(O,7),Circle(X,3))[1]; | ||
+ | M = midpoint(Q--R); | ||
+ | C = foot(P,Q,R); | ||
+ | D = foot(X,P,C); | ||
+ | fill(P--Q--R--cycle,yellow); | ||
+ | dot("$O$",O,S); | ||
+ | dot("$X$",X,N); | ||
+ | dot("$A$",A,SW); | ||
+ | dot("$B$",B,SE); | ||
+ | dot("$P$",P,S); | ||
+ | dot("$Q$",Q,E); | ||
+ | dot("$R$",R,N); | ||
+ | dot("$M$",M,dir(M)); | ||
+ | dot("$C$",C,NE); | ||
+ | dot("$D$",D,SE); | ||
+ | markscalefactor=0.0375; | ||
+ | draw(rightanglemark(O,M,R),red); | ||
+ | draw(rightanglemark(P,C,M),red); | ||
+ | draw(rightanglemark(P,D,X),red); | ||
+ | draw(rightanglemark(O,P,X),red); | ||
+ | draw(P--Q--R--cycle); | ||
+ | draw(arc(O, 7, 0, 180)^^A--B^^Circle(X,3)); | ||
+ | draw(O--M^^X--P); | ||
+ | draw(P--C^^X--D,dashed); | ||
+ | </asy> | ||
+ | Clearly, quadrilateral <math>XDCM</math> is a rectangle. Since <math>\angle XPD=\angle OXP</math> by alternate interior angles, we have <math>\triangle XPD\sim\triangle OXP</math> by the AA Similarity, with the ratio of similitude <math>\frac{XP}{OX}=\frac 35.</math> Therefore, we get <math>PD=\frac 95</math> and <math>PC=PD+DC=PD+XM=\frac 95 + \frac 32 = \frac{33}{10}.</math> | ||
The area of <math>\triangle PQR</math> is <cmath>\frac12\cdot QR\cdot PC=\frac12\cdot3\sqrt3\cdot\frac{33}{10}=\frac{99\sqrt3}{20},</cmath> from which the answer is <math>99+3+20=\boxed{\textbf{(D) } 122}.</math> | The area of <math>\triangle PQR</math> is <cmath>\frac12\cdot QR\cdot PC=\frac12\cdot3\sqrt3\cdot\frac{33}{10}=\frac{99\sqrt3}{20},</cmath> from which the answer is <math>99+3+20=\boxed{\textbf{(D) } 122}.</math> | ||
Line 26: | Line 82: | ||
==Solution 2 (Trigonometry)== | ==Solution 2 (Trigonometry)== | ||
<asy> | <asy> | ||
+ | size(150); | ||
draw(circle((7,0),7)); | draw(circle((7,0),7)); | ||
pair A = (0, 0); | pair A = (0, 0); | ||
Line 61: | Line 118: | ||
==Solution 3 (Weighted Averages and Similar Triangles)== | ==Solution 3 (Weighted Averages and Similar Triangles)== | ||
− | <asy> | + | <asy> size(300); pair C = (7, 0); draw(arc(C, 7, 0, 180)); pair A = (0, 0), B = (14, 0); draw(A--B); draw(circle((11,3),3)); label("$A$", A, SSE); label("$B$", B, SSW); label("$C$", (A+B)/2, S); label("$O$", (11, 3), E); label("$P$", (11, 0), S); pair O = (11, 3), P = (11, 0), Q = intersectionpoints(circle(C, 7), circle(O, 3))[1], R = intersectionpoints(circle(C, 7), circle(O, 3))[0], S = (Q+R)/2, N = (121/8, 0), T = (8/11)*N + (3/11)*R, X = (4/7)*T + (3/7)*S; draw(C--O, blue); draw(O--S, red); draw(C--Q); draw(C--R); draw(Q--N--B); draw(O--P); draw(O--Q); draw(O--R); draw(P--Q--R--cycle); draw(B--T); draw(P--X); label("$Q$", Q, NNE); label("$R$", R, E); label("$S$", S, ENE); label("$N$", N, SSE); label("$T$", T, ENE); label("$X$", X, NE); draw(rightanglemark(P, X, Q)); draw(rightanglemark(B, T, R)); draw(rightanglemark(C, S, Q)); </asy> |
Define points as shown above, where <math>N=\overleftrightarrow{PA}\cap\overleftrightarrow{QR}</math>. The area of <math>\triangle PQR</math> is simply<cmath>\dfrac{1}{2}PX\cdot QR=\dfrac{3\sqrt{3}}{2}PX;</cmath> | Define points as shown above, where <math>N=\overleftrightarrow{PA}\cap\overleftrightarrow{QR}</math>. The area of <math>\triangle PQR</math> is simply<cmath>\dfrac{1}{2}PX\cdot QR=\dfrac{3\sqrt{3}}{2}PX;</cmath> | ||
it remains to compute the value of <math>PX</math>. Note that <math>PX</math> is simply a weighted average of <math>BT</math> and <math>CS;</math> it is <math>\dfrac{CP}{BP}</math> times closer to <math>BT</math> than it is to <math>CS</math>. Observe that<cmath>CS=\sqrt{CQ^{2}-\left(\dfrac{1}{2}QR\right)^{2}}=\sqrt{7^{2}-\left(\dfrac{3\sqrt{3}}{2}\right)^{2}}=6.5</cmath> | it remains to compute the value of <math>PX</math>. Note that <math>PX</math> is simply a weighted average of <math>BT</math> and <math>CS;</math> it is <math>\dfrac{CP}{BP}</math> times closer to <math>BT</math> than it is to <math>CS</math>. Observe that<cmath>CS=\sqrt{CQ^{2}-\left(\dfrac{1}{2}QR\right)^{2}}=\sqrt{7^{2}-\left(\dfrac{3\sqrt{3}}{2}\right)^{2}}=6.5</cmath> | ||
Line 68: | Line 125: | ||
~sugar_rush | ~sugar_rush | ||
+ | |||
+ | ==Solution 4 (Similar Triangles)== | ||
+ | Let <math>O_{1}</math> be the center of <math>\odot\Gamma, O_2</math> be the center of <math>\odot\Omega,</math> and <math>M</math> be the midpoint of <math>\overline{QR}.</math> We have <math>O_{1}M=\sqrt{7^2-\left(\frac{3\sqrt3}{2}\right)^2}=\frac{13}{2}</math> and by Extended Law of Sines, the radius of <math>\odot\Omega</math> is <math>\frac{3\sqrt3}{2\sin 60^\circ}=3</math> so <math>O_{2}M=3\cos 60^\circ=\frac{3}{2}.</math> Therefore <math>O_{1}O_{2}=O_{1}M-O_{2}M=5</math> and <math>O_{1}P=\sqrt{5^2 - 3^2}=4.</math> | ||
+ | |||
+ | Let <math>X=\overline{AB}\cap\overline{QR}.</math> Obviously <cmath>\angle O_{1}PO_{2}=\angle O_{1}MX=90^\circ~\text{and}~\angle PO_{1}O_{2}=\angle MO_{1}X=\arcsin\left(\frac{3}{5}\right)</cmath> so <math>\triangle PO_{1}O_{2}\sim\triangle MO_{1}X</math> with ratio <math>\frac{PO_{1}}{MO_{1}}=\frac{4}{\tfrac{13}{2}}=\frac{8}{13}.</math> Therefore <math>O_1X=\frac{13}{8}\cdot O_{1}O_{2}=\frac{13}{8}\cdot5=\frac{65}{8}</math> and <math>MX=\frac{13}{8}\cdot PO_{2}=\frac{13}{8}\cdot3=\frac{39}{8}.</math> | ||
+ | |||
+ | Let <math>H</math> denote the foot of the altitude from <math>P</math> to <math>\overline{QR}.</math> Because <math>\overline{PH}\parallel\overline{O_{1}M},</math> it follows that <math>\triangle PHX\sim\triangle O_{1}MX.</math> This similarity has ratio <cmath>\frac{PX}{O_{1}X}=1-\frac{O_{1}P}{O_{1}X}=1-\frac{4}{\tfrac{65}{8}}=1-\frac{32}{65}=\frac{33}{65}.</cmath> We therefore have <math>PH=\frac{33}{65}\cdot O_{1}M=\frac{33}{65}\cdot\frac{13}{2}=\frac{33}{10}.</math> | ||
+ | |||
+ | Finally, the area of <math>\triangle PQR</math> is <cmath>\frac{1}{2}\cdot QR\cdot PH=\frac{1}{2}\cdot3\sqrt3\cdot\frac{33}{10}=\frac{99\sqrt3}{20},</cmath> so the answer is <math>99+3+20=\boxed{\textbf{(D)}~122}.</math> | ||
+ | |||
+ | ~inventivedant | ||
+ | |||
+ | ==Solution 5 (Law of Sine and Power of a Point)== | ||
+ | |||
+ | [[File:2021AMC12AP24.png|center|400px]] | ||
+ | |||
+ | By the Law of Sine in <math>\triangle PQR</math> and its circumcircle <math>\odot \Omega</math>, <math>2r_{\Omega} = \frac{QR}{ \sin 60^{\circ} } = \frac{ 3\sqrt{3} }{ \frac{ \sqrt{3} }{2} } = 6</math>, <math>r_{\Omega} = 3</math> | ||
+ | |||
+ | <cmath>\Gamma \Omega = \sqrt{r_{\Gamma}^2 - \left( \frac{ PQ }{2}\right)^2} - \sqrt{r_{\Omega}^2 - \left( \frac{PQ}{2}\right)^2} = \sqrt{7^2 - \left( \frac{ 3 \sqrt{3} }{2}\right)^2} - \sqrt{3^2 - \left( \frac{ 3 \sqrt{3} }{2}\right)^2} = \frac{13}{2} - \frac32 = 5, \quad \Gamma P = \sqrt{5^2 - 3^2} = 4</cmath> | ||
+ | |||
+ | By Power of a Point in <math>\odot \Gamma</math> , <math>PQ \cdot PS = PA \cdot PB = (7+4)(7-4) = 33</math>. | ||
+ | |||
+ | By the Law of Sine in <math>\triangle PRS</math>, <math>\frac{PR}{PS} = \frac{ \sin \angle PSR }{ \sin \angle PRS }</math> | ||
+ | |||
+ | By the Law of Sine in <math>\triangle QRS</math> and its circumcircle <math>\odot \Gamma</math>, <math>\frac{QR}{ \sin \angle PSR } = 14</math>, <math>\frac{ 3\sqrt{3} }{ \sin \angle PSR } = 14</math>, <math>\sin \angle PSR = \frac{ 3\sqrt{3} }{14}</math>, <math>\cos \angle PSR = \frac{ 13 }{14}</math> | ||
+ | |||
+ | <cmath>\sin \angle PRS = \sin ( 60^{\circ} - \angle PSR ) = \sin 60^{\circ} \cos \angle PSR - \sin \angle PSR \cos 60^{\circ} = \frac{ \sqrt{3} }{2} \cdot \frac{ 13 }{14} - \frac{ 3\sqrt{3} }{14} \cdot \frac12 = \frac{ 5\sqrt{3} }{14}</cmath> | ||
+ | |||
+ | <cmath>\frac{PR}{PS} = \frac{ \frac{ 3\sqrt{3} }{14} }{ \frac{ 5\sqrt{3} }{14} } = \frac35, \quad PQ \cdot PR = PQ \cdot PS \cdot \frac{PR}{PS} = 33 \cdot \frac35 = \frac{99}{5}</cmath> | ||
+ | |||
+ | <cmath>[PQR] = \frac12 \cdot \sin 60^{\circ} \cdot PQ \cdot PR = \frac12 \cdot \frac12 \cdot \frac{99}{5} = \frac{ 99\sqrt{3} }{20}, \quad 99 + 3 + 20 = \boxed{\textbf{(D)}~122}</cmath> | ||
+ | |||
+ | ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ||
+ | |||
+ | ==Solution 6 (Analytic Geometry)== | ||
+ | Following Solution 4, We have <math>O_{1}</math> (0,0) , <math>O_{2}</math> (4,3). | ||
+ | We can write the equation of the two circles as: <cmath>\odot\Gamma : x^{2}+y^{2}=7^{2}...(1)</cmath> | ||
+ | <cmath>\odot\Omega : (x-4)^{2}+(y-3)^{2}=3^{2}...(2)</cmath> | ||
+ | By substituting (1) into (2), we get <cmath>8x+6y-65=0...(3)</cmath> | ||
+ | Notice (3) is the relationship between <math>x</math> value and <math>y</math> value, in other words, (3) is the linear equation that go through <math>R</math> and <math>Q</math>. | ||
+ | Let the height drops from <math>P</math> to <math>QR</math> at <math>H</math>. Therefore, we have <cmath>Area \triangle PQR =\frac{1}{2}\cdot{QR}\cdot{PH}</cmath> | ||
+ | So <cmath>QR=3\sqrt{3}</cmath> | ||
+ | And by distance formula, <math>PH</math> is the distance from <math>P</math> (4,0) to <math>\overline{QR}</math>. <cmath>PH={\frac{|8\cdot4+6\cdot0-65|}{\sqrt{8^{2}+6^{2}}}=\frac{33}{10}}</cmath> | ||
+ | Thus, We get <cmath>Area \triangle PQR =\frac{1}{2}\cdot3\sqrt{3}\cdot\frac{33}{10}=\frac{99\sqrt{3}}{20}</cmath> | ||
+ | So the answer is <math>99+3+20=\boxed{\textbf{(D)}~122}.</math> | ||
+ | |||
+ | ~ERiccc | ||
+ | |||
+ | ==Video Solution by MOP 2024== | ||
+ | https://youtube.com/watch?v=UJ_M_cjul1Q | ||
+ | |||
+ | ~r00tsOfUnity | ||
==Video Solution by Punxsutawney Phil== | ==Video Solution by Punxsutawney Phil== | ||
https://youtube.com/watch?v=cEHF5iWMe9c | https://youtube.com/watch?v=cEHF5iWMe9c | ||
+ | |||
+ | This is invalid. ~r00tsOfUnity | ||
== Video Solution by OmegaLearn (Similar Triangles, Law of Sines, Law of Cosines ) == | == Video Solution by OmegaLearn (Similar Triangles, Law of Sines, Law of Cosines ) == | ||
Line 76: | Line 187: | ||
~pi_is_3.14 | ~pi_is_3.14 | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/VUvSH-8AxzM | ||
+ | |||
+ | ~MathProblemSolvingSkills.com | ||
==See also== | ==See also== |
Latest revision as of 15:24, 15 June 2024
Contents
- 1 Problem
- 2 Diagram
- 3 Solution 1 (Possible Without Trigonometry)
- 4 Solution 2 (Trigonometry)
- 5 Solution 3 (Weighted Averages and Similar Triangles)
- 6 Solution 4 (Similar Triangles)
- 7 Solution 5 (Law of Sine and Power of a Point)
- 8 Solution 6 (Analytic Geometry)
- 9 Video Solution by MOP 2024
- 10 Video Solution by Punxsutawney Phil
- 11 Video Solution by OmegaLearn (Similar Triangles, Law of Sines, Law of Cosines )
- 12 Video Solution
- 13 See also
Problem
Semicircle has diameter of length . Circle lies tangent to at a point and intersects at points and . If and , then the area of equals , where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. What is ?
Diagram
~MRENTHUSIASM
Solution 1 (Possible Without Trigonometry)
Let be the center of the semicircle and be the center of the circle.
Applying the Extended Law of Sines to we find the radius of Alternatively, by the Inscribed Angle Theorem, is a triangle with base Dividing into two congruent triangles, we get that the radius of is by the side-length ratios.
Let be the midpoint of By the Perpendicular Chord Bisector Converse, we have and Together, points and must be collinear.
By the SAS Congruence, we have both of which are triangles. By the side-length ratios, we obtain and By the Pythagorean Theorem on right we get and By the Pythagorean Theorem on right we get
Let be the foot of the perpendicular from to and be the foot of the perpendicular from to as shown below: Clearly, quadrilateral is a rectangle. Since by alternate interior angles, we have by the AA Similarity, with the ratio of similitude Therefore, we get and
The area of is from which the answer is
~MRENTHUSIASM
Solution 2 (Trigonometry)
Suppose we label the points as shown in the diagram above, where is the center of the semicircle and is the center of the circle tangent to . Since , we have and is a triangle, which can be split into two triangles by the altitude from . Since we know by triangles. The area of this part of is . We would like to add this value to the sum of the areas of the other two parts of .
To find the areas of the other two parts of using the area formula, we need the sides and included angles. Here we know the sides but what we don't know are the angles. So it seems like we will have to use an angle from another triangle and combine them with the angles we already know to find these angles easily. We know that and triangles and are congruent as they share a side, and . Therefore . Suppose . Then , and since , this simplifies to . This factors nicely as , so as can't be . Since and , we now know that is a right triangle. This may be useful info for later as we might use an angle in this triangle to find the areas of the other two parts of .
Let . Then and . The sum of the areas of and is which we will add to to get the area of . Observe that and similarly . Adding these two gives and multiplying that by gets us which we add to to get . The answer is
~sugar_rush
Solution 3 (Weighted Averages and Similar Triangles)
Define points as shown above, where . The area of is simply it remains to compute the value of . Note that is simply a weighted average of and it is times closer to than it is to . Observe that since the radius of is as its diameter is . Note also by the Extended Law of Sines the radius of is so . Since and are collinear by symmetry we have so and . Therefore, is a right triangle; since and . Therefore so . Since we have . Therefore so is times as close to as to we can compute . The area of isand .
~sugar_rush
Solution 4 (Similar Triangles)
Let be the center of be the center of and be the midpoint of We have and by Extended Law of Sines, the radius of is so Therefore and
Let Obviously so with ratio Therefore and
Let denote the foot of the altitude from to Because it follows that This similarity has ratio We therefore have
Finally, the area of is so the answer is
~inventivedant
Solution 5 (Law of Sine and Power of a Point)
By the Law of Sine in and its circumcircle , ,
By Power of a Point in , .
By the Law of Sine in ,
By the Law of Sine in and its circumcircle , , , ,
Solution 6 (Analytic Geometry)
Following Solution 4, We have (0,0) , (4,3). We can write the equation of the two circles as: By substituting (1) into (2), we get Notice (3) is the relationship between value and value, in other words, (3) is the linear equation that go through and . Let the height drops from to at . Therefore, we have So And by distance formula, is the distance from (4,0) to . Thus, We get So the answer is
~ERiccc
Video Solution by MOP 2024
https://youtube.com/watch?v=UJ_M_cjul1Q
~r00tsOfUnity
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=cEHF5iWMe9c
This is invalid. ~r00tsOfUnity
Video Solution by OmegaLearn (Similar Triangles, Law of Sines, Law of Cosines )
~pi_is_3.14
Video Solution
~MathProblemSolvingSkills.com
See also
2021 AMC 12A (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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.