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Difference between revisions of "2017 AMC 8 Problems/Problem 18"

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==Solution==
 
==Solution==
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We can see a Pythagorean triple's two  longer lengths: <math>12</math>, <math>13</math>. So <math>BD</math> should be <math>5</math>. This is certainly the case because <math>3^2 + 4^2 = 5^2</math>, which is <math>BD</math>. Thus the area of quadrialteral <math>ABCD</math> is <math>30-6 = \boxed{\textbf{(B)}\ 24}.</math>
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We first connect point <math>B</math> with point <math>D</math>. We can see that it is a 3-4-5 right triangle. Using Pythagorean triples, we see that <math>\triangle BDA</math> is a right triangle [I NEED SOMEBODY TO PROVE THIS] , therefore <math>\overline{DA}</math> is <math>13</math>. With these lengths, we can solve the problem. The area of <math>\triangle BDA</math> is <math>\frac{5\cdot 12}{2}</math>, and the area of the smaller 3-4-5 triangle is <math>\frac{3\cdot 4}{2}</math>. Thus, the area of quadrialteral <math>ABCD</math> is <math>30-6 = \boxed{\textbf{(B)}\ 24}.</math>
  
 
==See Also==
 
==See Also==

Revision as of 13:02, 5 January 2018

Problem 18

In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. [asy]draw((0,0)--(2.4,3.6)--(0,5)--(12,0)--(0,0)); label("$B$", (0, 0), SW); label("$A$", (12, 0), ESE); label("$C$", (2.4, 3.6), SE); label("$D$", (0, 5), N);[/asy] What is the area of quadrilateral $ABCD$?

$\textbf{(A) }12\qquad\textbf{(B) }24\qquad\textbf{(C) }26\qquad\textbf{(D) }30\qquad\textbf{(E) }36$

Solution

We first connect point $B$ with point $D$. We can see that it is a 3-4-5 right triangle. Using Pythagorean triples, we see that $\triangle BDA$ is a right triangle [I NEED SOMEBODY TO PROVE THIS] , therefore $\overline{DA}$ is $13$. With these lengths, we can solve the problem. The area of $\triangle BDA$ is $\frac{5\cdot 12}{2}$, and the area of the smaller 3-4-5 triangle is $\frac{3\cdot 4}{2}$. Thus, the area of quadrialteral $ABCD$ is $30-6 = \boxed{\textbf{(B)}\ 24}.$

See Also

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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 AJHSME/AMC 8 Problems and Solutions

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