Difference between revisions of "2017 AMC 8 Problems/Problem 18"
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==Solution 2== | ==Solution 2== | ||
− | <math>\triangle BCD</math> is a 3-4-5 right triangle. So the area of <math>\triangle BCD</math> is 6. Then we can use Heron's formula to compute the area of <math>\triangle ABD</math> whose sides have lengths 5,12,and 13. The area of <math>\triangle ABD</math> = <math>\sqrt{s(s-5)(s-12)(s-13)}</math> , where s is the semi-perimeter of the triangle, that is <math>s=(5+12+13)/2=15</math> | + | <math>\triangle BCD</math> is a 3-4-5 right triangle. So the area of <math>\triangle BCD</math> is 6. Then we can use Heron's formula to compute the area of <math>\triangle ABD</math> whose sides have lengths 5,12,and 13. The area of <math>\triangle ABD</math> = <math>\sqrt{s(s-5)(s-12)(s-13)}</math> , where s is the semi-perimeter of the triangle, that is <math>s=(5+12+13)/2=15.</math> Thus, the area of <math>\triangle ABD</math> =30, so the area of <math>ABCD</math> is <math>30-6 = \boxed{\textbf{(B)}\ 24}.</math> ---LarryFlora |
==See Also== | ==See Also== |
Revision as of 14:48, 17 June 2021
Contents
Problem
In the non-convex quadrilateral shown below,
is a right angle,
,
,
, and
. What is the area of quadrilateral
?
Solution 1
We first connect point with point
.
We can see that is a 3-4-5 right triangle. We can also see that
is a right triangle, by the 5-12-13 Pythagorean triple. With these lengths, we can solve the problem. The area of
is
, and the area of the smaller 3-4-5 triangle is
. Thus, the area of quadrialteral
is
Solution 2
is a 3-4-5 right triangle. So the area of
is 6. Then we can use Heron's formula to compute the area of
whose sides have lengths 5,12,and 13. The area of
=
, where s is the semi-perimeter of the triangle, that is
Thus, the area of
=30, so the area of
is
---LarryFlora
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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