Difference between revisions of "2002 AMC 12P Problems/Problem 24"
The 76923th (talk | contribs) m (→Solution) |
The 76923th (talk | contribs) m (→Solution) |
||
(8 intermediate revisions by the same user not shown) | |||
Line 21: | Line 21: | ||
Consider triangles <math>EPS</math>, <math>EQT</math>, and <math>ERU</math>. Each of these triangles have a right angle and an angle equal to the dihedral angle of the tetrahedron, so they are all similar by AA similarity. In particular, we know that <math>\frac{EP}{ES} = \frac{EQ}{ET} = \frac{ER}{EU} = \frac{EP+EQ+ER}{ES+ET+EU} = \frac{s}{S}</math>. | Consider triangles <math>EPS</math>, <math>EQT</math>, and <math>ERU</math>. Each of these triangles have a right angle and an angle equal to the dihedral angle of the tetrahedron, so they are all similar by AA similarity. In particular, we know that <math>\frac{EP}{ES} = \frac{EQ}{ET} = \frac{ER}{EU} = \frac{EP+EQ+ER}{ES+ET+EU} = \frac{s}{S}</math>. | ||
− | It remains to find <math>\frac{EP}{ES}</math>, or equivalently, <math>cos(\angle DSE)</math>. | + | It remains to find <math>\frac{EP}{ES}</math>, or equivalently, <math>\sin(\angle DSE)</math>. |
+ | |||
+ | We know <math>SE = \frac{1}{3}DS</math> by the centroid property. Therefore, <math>\cos(\angle DSE) = \frac{1}{3}</math>, so <math>\sin(\angle DSE) = \sqrt{1-(\frac{1}{3})^2} = \boxed {\text{(B) }\frac{2 \sqrt{2}}{3}}</math>. | ||
== See also == | == See also == | ||
{{AMC12 box|year=2002|ab=P|num-b=23|num-a=25}} | {{AMC12 box|year=2002|ab=P|num-b=23|num-a=25}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 18:04, 10 March 2024
Problem
Let be a regular tetrahedron and Let be a point inside the face Denote by the sum of the distances from to the faces and by the sum of the distances from to the edges Then equals
Solution
Assume points , , and are on faces , , and respectively such that , , and .
Assume points , , and are on edges , , and respectively such that , , and .
Consider triangles , , and . Each of these triangles have a right angle and an angle equal to the dihedral angle of the tetrahedron, so they are all similar by AA similarity. In particular, we know that .
It remains to find , or equivalently, .
We know by the centroid property. Therefore, , so .
See also
2002 AMC 12P (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.