Difference between revisions of "2016 AMC 12B Problems/Problem 10"
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==Solution== | ==Solution== | ||
− | By distance formula we have <math>(a-b)^2+(b-a)^2*2*(a+b)^2 = 256</math>. | + | Note that the slope of <math>PQ</math> is <math>\frac{a-b}{b-a}=-1</math> and the slope of <math>PS</math> is <math>\frac{b+a}{a+b}=1</math>. Hence, <math>PQ\perp PS</math> and we can similarly prove that the other angles are right angles. This means that <math>PQRS</math> is a rectangle. By distance formula we have <math>(a-b)^2+(b-a)^2*2*(a+b)^2 = 256</math>. Simplifying we get <math>(a-b)(a+b) = 8</math>. Thus <math>a+b</math> and <math>a-b</math> have to be a factor of 8. The only way for them to be factors of <math>8</math> and remain integers is if <math>a+b = 4</math> and <math>a-b = 2</math>. So the answer is <math>\boxed{\textbf{(A)}\ 4}</math> |
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Solution by I_Dont_Do_Math | Solution by I_Dont_Do_Math | ||
+ | |||
+ | ==Solution 2== | ||
+ | Solution by e_power_pi_times_i | ||
+ | |||
+ | |||
+ | By the Shoelace Theorem, the area of the quadrilateral is <math>2a^2 - 2b^2</math>, so <math>a^2 - b^2 = 8</math>. Since <math>a</math> and <math>b</math> are integers, <math>a = 3</math> and <math>b = 1</math>, so <math>a + b = \boxed{\textbf{(A)}\ 4}</math>. | ||
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+ | ==Video Solution by TheBeautyofMath== | ||
+ | https://www.youtube.com/watch?v=Eq2A2TTahqU | ||
+ | with a second example of Shoelace Theorem done after this problem | ||
+ | |||
+ | ~IceMatrix | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2016|ab=B|num-b=9|num-a=11}} | {{AMC12 box|year=2016|ab=B|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 06:08, 19 February 2021
Problem
A quadrilateral has vertices , , , and , where and are integers with . The area of is . What is ?
Solution
Note that the slope of is and the slope of is . Hence, and we can similarly prove that the other angles are right angles. This means that is a rectangle. By distance formula we have . Simplifying we get . Thus and have to be a factor of 8. The only way for them to be factors of and remain integers is if and . So the answer is
Solution by I_Dont_Do_Math
Solution 2
Solution by e_power_pi_times_i
By the Shoelace Theorem, the area of the quadrilateral is , so . Since and are integers, and , so .
Video Solution by TheBeautyofMath
https://www.youtube.com/watch?v=Eq2A2TTahqU with a second example of Shoelace Theorem done after this problem
~IceMatrix
See Also
2016 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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 12 Problems and Solutions |
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