Difference between revisions of "2017 AMC 10A Problems/Problem 17"
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Because <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> are lattice points, there are only a few coordinates that actually satisfy the equation. The coordinates are <math>(\pm 3,\pm 4), (\pm 4, \pm 3), (0,\pm 5),</math> and <math>(\pm 5,0).</math> We want to maximize <math>PQ</math> and minimize <math>RS.</math> They also have to be the square root of something, because they are both irrational. The greatest value of <math>PQ</math> happens when <math>P</math> and <math>Q</math> are almost directly across from each other and are in different quadrants. For example, the endpoints of the segment could be <math>(-4,3)</math> and <math>(3,-4)</math> because the two points are almost across from each other. Another possible pair could be <math>(-4,3)</math> and <math>(5,0)</math>. To find out which segment is longer, we have to compare the distances from their endpoints to a diameter (which must be the longest possible segment). The closest diameter would be from <math>(-4,3)</math> to <math>(4,-3)</math>. The distance between <math>(3,-4)</math> and <math>(4,-3)</math> is shorter than the distance between <math>(5,0)</math> and <math>(4,-3)</math>. Therefore, the segment from <math>(-4,3)</math> to <math>(3,-4)</math> is the longest attainable. The least value of <math>RS</math> is when the two endpoints are in the same quadrant and are very close to each other. This can occur when, for example, <math>R</math> is <math>(3,4)</math> and <math>S</math> is <math>(4,3).</math> They are in the same quadrant and no other point on the circle with integer coordinates is closer to the point <math>(3,4)</math> than <math>(4,3).</math> Using the distance formula, we get that <math>PQ</math> is <math>\sqrt{98}</math> and that <math>RS</math> is <math>\sqrt{2}.</math> <math>\frac{\sqrt{98}}{\sqrt{2}}=\sqrt{49}=\boxed{\mathrm{\textbf{(D)}}\ 7}</math> | Because <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> are lattice points, there are only a few coordinates that actually satisfy the equation. The coordinates are <math>(\pm 3,\pm 4), (\pm 4, \pm 3), (0,\pm 5),</math> and <math>(\pm 5,0).</math> We want to maximize <math>PQ</math> and minimize <math>RS.</math> They also have to be the square root of something, because they are both irrational. The greatest value of <math>PQ</math> happens when <math>P</math> and <math>Q</math> are almost directly across from each other and are in different quadrants. For example, the endpoints of the segment could be <math>(-4,3)</math> and <math>(3,-4)</math> because the two points are almost across from each other. Another possible pair could be <math>(-4,3)</math> and <math>(5,0)</math>. To find out which segment is longer, we have to compare the distances from their endpoints to a diameter (which must be the longest possible segment). The closest diameter would be from <math>(-4,3)</math> to <math>(4,-3)</math>. The distance between <math>(3,-4)</math> and <math>(4,-3)</math> is shorter than the distance between <math>(5,0)</math> and <math>(4,-3)</math>. Therefore, the segment from <math>(-4,3)</math> to <math>(3,-4)</math> is the longest attainable. The least value of <math>RS</math> is when the two endpoints are in the same quadrant and are very close to each other. This can occur when, for example, <math>R</math> is <math>(3,4)</math> and <math>S</math> is <math>(4,3).</math> They are in the same quadrant and no other point on the circle with integer coordinates is closer to the point <math>(3,4)</math> than <math>(4,3).</math> Using the distance formula, we get that <math>PQ</math> is <math>\sqrt{98}</math> and that <math>RS</math> is <math>\sqrt{2}.</math> <math>\frac{\sqrt{98}}{\sqrt{2}}=\sqrt{49}=\boxed{\mathrm{\textbf{(D)}}\ 7}</math> | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/umr2Aj9ViOA | ||
==See Also== | ==See Also== |
Revision as of 23:21, 20 January 2020
Contents
Problem
Distinct points ,
,
,
lie on the circle
and have integer coordinates. The distances
and
are irrational numbers. What is the greatest possible value of the ratio
?
Solution
Because ,
,
, and
are lattice points, there are only a few coordinates that actually satisfy the equation. The coordinates are
and
We want to maximize
and minimize
They also have to be the square root of something, because they are both irrational. The greatest value of
happens when
and
are almost directly across from each other and are in different quadrants. For example, the endpoints of the segment could be
and
because the two points are almost across from each other. Another possible pair could be
and
. To find out which segment is longer, we have to compare the distances from their endpoints to a diameter (which must be the longest possible segment). The closest diameter would be from
to
. The distance between
and
is shorter than the distance between
and
. Therefore, the segment from
to
is the longest attainable. The least value of
is when the two endpoints are in the same quadrant and are very close to each other. This can occur when, for example,
is
and
is
They are in the same quadrant and no other point on the circle with integer coordinates is closer to the point
than
Using the distance formula, we get that
is
and that
is
Video Solution
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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 10 Problems and Solutions |
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