Difference between revisions of "2021 AIME I Problems/Problem 6"
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==Problem== | ==Problem== | ||
− | Segments <math>\overline{AB}, \overline{AC},</math> and <math>\overline{AD}</math> are edges of a cube and <math>\overline{AG}</math> is a diagonal through the center of the cube. Point <math>P</math> satisfies <math> | + | Segments <math>\overline{AB}, \overline{AC},</math> and <math>\overline{AD}</math> are edges of a cube and <math>\overline{AG}</math> is a diagonal through the center of the cube. Point <math>P</math> satisfies <math>BP=60\sqrt{10}</math>, <math>CP=60\sqrt{5}</math>, <math>DP=120\sqrt{2}</math>, and <math>GP=36\sqrt{7}</math>. Find <math>AP.</math> |
==Solution 1== | ==Solution 1== |
Revision as of 11:55, 11 October 2022
Contents
[hide]Problem
Segments and are edges of a cube and is a diagonal through the center of the cube. Point satisfies , , , and . Find
Solution 1
First scale down the whole cube by . Let point have coordinates , point have coordinates , and be the side length. Then we have the equations These simplify into Adding the first three equations together, we get . Subtracting this from the fourth equation, we get , so . This means . However, we scaled down everything by so our answer is .
~JHawk0224
Solution 2 (Solution 1 with Slight Simplification)
Once the equations for the distance between point P and the vertices of the cube have been written. We can add the first, second, and third to receive, Subtracting the fourth equation gives Since point , and since we scaled the answer is .
~Aaryabhatta1
Solution 3
Let be the vertex of the cube such that is a square. By the British Flag Theorem, we can easily we can show that and Hence, adding the two equations together, we get . Substituting in the values we know, we get .
Thus, we can solve for , which ends up being .
Solution 4
For all points in space, define the function by . Then is linear; let be the center of . Then since is linear, where denotes the side length of the cube. Thus
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=vaRfI0l4s_8
See Also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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