Difference between revisions of "2012 AMC 10B Problems/Problem 12"
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== Problem == | == 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 <math>10\sqrt 2</math>, and <math>\angle BAC= 45^\circ</math>. Point D is 20 meters due north of point C. The distance AD is between which two integers? | + | Point <math>B</math> is due east of point <math>A</math>. Point <math>C</math> is due north of point <math>B</math>. The distance between points <math>A</math> and <math>C</math> is <math>10\sqrt 2</math>, and <math>\angle BAC = 45^\circ</math>. Point <math>D</math> is <math>20</math> meters due north of point <math>C</math>. The distance <math>AD</math> is between which two integers? |
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<math>\textbf{(A)}\ 30\ \text{and}\ 31 \qquad\textbf{(B)}\ 31\ \text{and}\ 32 \qquad\textbf{(C)}\ 32\ \text{and}\ 33 \qquad\textbf{(D)}\ 33\ \text{and}\ 34 \qquad\textbf{(E)}\ 34\ \text{and}\ 35</math> | <math>\textbf{(A)}\ 30\ \text{and}\ 31 \qquad\textbf{(B)}\ 31\ \text{and}\ 32 \qquad\textbf{(C)}\ 32\ \text{and}\ 33 \qquad\textbf{(D)}\ 33\ \text{and}\ 34 \qquad\textbf{(E)}\ 34\ \text{and}\ 35</math> | ||
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== Solution == | == Solution == | ||
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{{AMC10 box|year=2012|ab=B|num-b=11|num-a=13}} | {{AMC10 box|year=2012|ab=B|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
+ | [[Category: Introductory Geometry Problems]] |
Latest revision as of 20:22, 3 September 2021
Problem
Point is due east of point . Point is due north of point . The distance between points and is , and . Point is meters due north of point . The distance 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 |
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