Difference between revisions of "2010 AMC 10A Problems/Problem 7"
Mariekitty (talk | contribs) (→Solution) |
(→Solution) |
||
| Line 15: | Line 15: | ||
==Solution== | ==Solution== | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| Line 28: | Line 20: | ||
| + | <asy> | ||
| + | import olympiad; | ||
| + | draw((0,0)--(0,1)); | ||
| + | draw((0,1)--(0,1.7071067811865476), dotted); | ||
| + | draw((0,1)--(0.7071067811865476, 1.7071067811865476)); | ||
| + | draw((0.7071067811865476, 1.7071067811865476)--(1.4142135623730951,1)); | ||
| + | draw(anglemark((0.7071067811865476, 1.7071067811865476),(0,1),(0,1.7071067811865476))); | ||
| + | label("$45^{\circ}$", (0,1.25), NE); | ||
| + | draw((0, 1)--(1.4142135623730951,1), dotted); | ||
| + | draw((1.4142135623730951,1)--(0,0), green); | ||
| + | </asy> | ||
== See Also == | == See Also == | ||
{{AMC10 box|year=2010|ab=A|num-b=6|num-a=8}} | {{AMC10 box|year=2010|ab=A|num-b=6|num-a=8}} | ||
Revision as of 13:13, 2 August 2012
Problem 7
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?
Solution
Crystal first runs North for one mile. Changing directions, she runs Northeast for another mile. The angle difference between North and Northeast is 45 degrees. She then switches directions to Southeast, meaning a 90 degree angle change. The distance now from travelling North for one mile, and her current destination is 
miles, because it is the hypotenuse of a 45-45-90 triangle with side length one (mile). Therefore, Crystal's distance from her starting position, x, is equal to 
, which is equal to 
. The answer is 
![[asy] import olympiad; draw((0,0)--(0,1)); draw((0,1)--(0,1.7071067811865476), dotted); draw((0,1)--(0.7071067811865476, 1.7071067811865476)); draw((0.7071067811865476, 1.7071067811865476)--(1.4142135623730951,1)); draw(anglemark((0.7071067811865476, 1.7071067811865476),(0,1),(0,1.7071067811865476))); label("$45^{\circ}$", (0,1.25), NE); draw((0, 1)--(1.4142135623730951,1), dotted); draw((1.4142135623730951,1)--(0,0), green); [/asy]](http://latex.artofproblemsolving.com/4/f/8/4f85e07c71fb200daeb9e875a79276f9b0c38b4a.png)
See Also
| 2010 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 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 | ||
