Difference between revisions of "2022 AMC 12A Problems/Problem 12"
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Let <math>O</math> be the center of <math>\triangle ABD,</math> so <math>\overline{CO}\perp\overline{MOD}.</math> Note that <math>MO=\frac13 MD=\frac{\sqrt{3}}{3}.</math> | Let <math>O</math> be the center of <math>\triangle ABD,</math> so <math>\overline{CO}\perp\overline{MOD}.</math> Note that <math>MO=\frac13 MD=\frac{\sqrt{3}}{3}.</math> | ||
− | In <math>\triangle CMO,</math> we have <cmath>\cos(\angle CMD)=\frac{MO}{MC}=\boxed{\textbf{(B) } \frac13}.</cmath> | + | In right <math>\triangle CMO,</math> we have <cmath>\cos(\angle CMD)=\frac{MO}{MC}=\boxed{\textbf{(B) } \frac13}.</cmath> |
~MRENTHUSIASM | ~MRENTHUSIASM | ||
==Solution 2 (Law of Cosines)== | ==Solution 2 (Law of Cosines)== | ||
− | Without loss of generality, let the edge-length of <math>ABCD</math> be <math>2.</math> It follows that <math>CM = DM = \ | + | Without loss of generality, let the edge-length of <math>ABCD</math> be <math>2.</math> It follows that <math>CM=DM=\sqrt3.</math> |
By the Law of Cosines, <cmath>\cos(\angle CMD) = \frac{CM^2 + DM^2 - CD^2}{2(CM)(DM)} = \boxed{\textbf{(B) } \frac13}.</cmath> | By the Law of Cosines, <cmath>\cos(\angle CMD) = \frac{CM^2 + DM^2 - CD^2}{2(CM)(DM)} = \boxed{\textbf{(B) } \frac13}.</cmath> | ||
~jamesl123456 | ~jamesl123456 | ||
− | ==Solution 3 ( | + | ==Solution 3 (Double Angle Identities)== |
− | As done above, let the edge-length equal <math>2</math> (usually better than <math>1</math> because we can avoid fractions when dropping altitudes). Notice that the triangle stated in the question has two side-lengths that are the altitudes of two equilateral triangles. By dropping the equilateral triangles' altitude and using <math>30</math>-<math>60</math>-<math>90</math> properties, we find that the other two sides are equal to <math>\sqrt{3}</math>. Now by dropping the main triangle's altitude, we see it equals <math>\sqrt{2}</math> from the Pythagorean Theorem. we can use the Double Angle Identities for Cosine. Doing this, we obtain <cmath>\cos(\angle CMD) = \frac{2}{3} - \frac13 = \boxed{\textbf{(B) } \frac13}.</cmath> | + | As done above, let the edge-length equal <math>2</math> (usually better than <math>1</math> because we can avoid fractions when dropping altitudes). Notice that the triangle stated in the question has two side-lengths that are the altitudes of two equilateral triangles. By dropping the equilateral triangles' altitude and using <math>30^{\circ}</math>-<math>60^{\circ}</math>-<math>90^{\circ}</math> properties, we find that the other two sides are equal to <math>\sqrt{3}</math>. Now by dropping the main triangle's altitude, we see it equals <math>\sqrt{2}</math> from the Pythagorean Theorem. we can use the Double Angle Identities for Cosine. Doing this, we obtain <cmath>\cos(\angle CMD) = \frac{2}{3} - \frac13 = \boxed{\textbf{(B) } \frac13}.</cmath> |
− | |||
~Misclicked | ~Misclicked | ||
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~Education, the Study of Everything | ~Education, the Study of Everything | ||
+ | |||
+ | ==Video Solution 1 (Smart and Simple)== | ||
+ | https://youtu.be/7yAh4MtJ8a8?si=9uWHOngb2PTMRpg8&t=2423 | ||
+ | |||
+ | ~Math-X | ||
== See Also == | == See Also == |
Latest revision as of 22:42, 25 October 2023
Contents
[hide]Problem
Let be the midpoint of in regular tetrahedron . What is ?
Diagram
~MRENTHUSIASM
Solution 1 (Right Triangles)
Without loss of generality, let the edge-length of be It follows that
Let be the center of so Note that
In right we have ~MRENTHUSIASM
Solution 2 (Law of Cosines)
Without loss of generality, let the edge-length of be It follows that
By the Law of Cosines, ~jamesl123456
Solution 3 (Double Angle Identities)
As done above, let the edge-length equal (usually better than because we can avoid fractions when dropping altitudes). Notice that the triangle stated in the question has two side-lengths that are the altitudes of two equilateral triangles. By dropping the equilateral triangles' altitude and using -- properties, we find that the other two sides are equal to . Now by dropping the main triangle's altitude, we see it equals from the Pythagorean Theorem. we can use the Double Angle Identities for Cosine. Doing this, we obtain ~Misclicked
Video Solution 1 (Quick and Simple)
~Education, the Study of Everything
Video Solution 1 (Smart and Simple)
https://youtu.be/7yAh4MtJ8a8?si=9uWHOngb2PTMRpg8&t=2423
~Math-X
See Also
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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.