Difference between revisions of "2012 AMC 10B Problems/Problem 12"
(→Problem) |
(→Solution) |
||
Line 9: | Line 9: | ||
== Solution == | == Solution == | ||
+ | <asy> | ||
+ | unitsize(36); | ||
+ | pair A=(0,0); | ||
+ | pair B=(0,10); | ||
+ | pair C=(10,10); | ||
+ | pair D=(30,10); | ||
+ | draw(A--B); | ||
+ | </asy> | ||
If point B is due east of point A and point C is due north of point B, <math>\angle CBA</math> is a right angle. And if <math>\angle BAC = 45^\circ</math>, <math>\triangle CBA</math> is a 45-45-90 triangle. Thus, the lengths of sides <math>CB</math>, <math>BA</math>, and <math>AC</math> are in the ratio <math>1:1:\sqrt 2</math>, and <math>CB</math> is <math>10 \sqrt 2 \div \sqrt 2 = 10</math>. | If point B is due east of point A and point C is due north of point B, <math>\angle CBA</math> is a right angle. And if <math>\angle BAC = 45^\circ</math>, <math>\triangle CBA</math> is a 45-45-90 triangle. Thus, the lengths of sides <math>CB</math>, <math>BA</math>, and <math>AC</math> are in the ratio <math>1:1:\sqrt 2</math>, and <math>CB</math> is <math>10 \sqrt 2 \div \sqrt 2 = 10</math>. | ||
Revision as of 13:01, 31 December 2019
Problem
Point B is due east of point A. Point C is due north of point B. The distance between points A and C is , and
. Point D is 20 meters due north of point C. The distance AD is between which two integers?
Solution
If point B is due east of point A and point C is due north of point B,
is a right angle. And if
,
is a 45-45-90 triangle. Thus, the lengths of sides
,
, and
are in the ratio
, and
is
.
is clearly a right triangle with
on the side
.
is 20, so
.
By the Pythagorean Theorem, .
, and
. Thus,
must be between
and
. The answer is
.
See Also
2012 AMC 10B (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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.