Difference between revisions of "2021 AIME I Problems/Problem 6"
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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, <cmath>2(x^2 + y^2 + z^2) + (s-x)^2 + (s-y)^2 + (s-z)^2 = 250 + 125 + 200.</cmath> Subtracting the fourth equation gives, <cmath>2(x^2 + y^2 + z^2) = 575 - 63</cmath> <cmath>x^2 + y^2 + z^2 = 256</cmath> <cmath>\sqrt{x^2 + y^2 + z^2} = 16.</cmath> Since point <math>A = (0,0,0), PA = 16</math>, and since we scaled the answer is <math>16 \cdot 12 = \boxed{192}</math> | 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, <cmath>2(x^2 + y^2 + z^2) + (s-x)^2 + (s-y)^2 + (s-z)^2 = 250 + 125 + 200.</cmath> Subtracting the fourth equation gives, <cmath>2(x^2 + y^2 + z^2) = 575 - 63</cmath> <cmath>x^2 + y^2 + z^2 = 256</cmath> <cmath>\sqrt{x^2 + y^2 + z^2} = 16.</cmath> Since point <math>A = (0,0,0), PA = 16</math>, and since we scaled the answer is <math>16 \cdot 12 = \boxed{192}</math> | ||
~Aaryabhatta1 | ~Aaryabhatta1 | ||
− | ==See | + | ==Solution 3== |
+ | |||
+ | By Pythagorean Theorem, easly we can show that | ||
+ | <cmath>PA^2 + PE^2 = PB^2 + PD^2</cmath> | ||
+ | <cmath>PA^2 + PG^2 = PC^2 + PE^2</cmath> | ||
+ | Hence, <math>2PA^2 + PG^2 = PB^2 + PC^2 + PD^2</math>. <math>2PA^2 + 7\cdot 36^2 =10\cdot60^2 + 5\cdot 60^2 + 2\cdot 120^2 </math>. | ||
+ | Thus <math>PA</math> is <math>\boxed{192}</math>. | ||
+ | |||
+ | ==See Also== | ||
{{AIME box|year=2021|n=I|num-b=5|num-a=7}} | {{AIME box|year=2021|n=I|num-b=5|num-a=7}} | ||
Revision as of 05:59, 13 March 2021
Contents
Problem
Segments and are edges of a cube and is a diagonal through the center of the cube. Point satisfies and . What is ?
Solution 1
First scale down the whole cube by 12. Let point P have coordinates , A 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 12 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
By Pythagorean Theorem, easly we can show that Hence, . . Thus is .
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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