Difference between revisions of "2019 AMC 8 Problems/Problem 4"
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Which means area = <math>\frac{d_1*d_2}{2}</math> = <math>\frac{24*10}{2}</math> = <math>120</math> | Which means area = <math>\frac{d_1*d_2}{2}</math> = <math>\frac{24*10}{2}</math> = <math>120</math> | ||
| − | <math>\boxed{\textbf{(D)}\ 120}</math> | + | <math>\boxed{\textbf{(D)}\ 120}</math> ~phoenixfire |
==See Also== | ==See Also== | ||
Revision as of 01:27, 20 November 2019
Solution
![[asy] draw((-12,0)--(0,5)); draw((0,5)--(12,0)); draw((12,0)--(0,-5)); draw((0,-5)--(-12,0)); draw((0,0)--(12,0)); draw((0,0)--(0,5)); draw((0,0)--(-12,0)); draw((0,0)--(0,-5)); dot((-12,0)); dot((0,5)); dot((12,0)); dot((0,-5)); label("A",(-12,0),W); label("B",(0,5),N); label("C",(12,0),E); label("D",(0,-5),S); label("E",(0,0),SW); [/asy]](http://latex.artofproblemsolving.com/2/3/0/230af487f6bb84b2940aef916f01294a89dfc2cf.png)
Because it is a rhombus all sides are equal. Implies all sides are 13. In a rhombus diagonals are perpendicular and bisect each other.
Which means 
= 
= 
.
Consider one of the right triangles.
![[asy] draw((-12,0)--(0,5)); draw((0,0)--(-12,0)); draw((0,0)--(0,5)); dot((-12,0)); dot((0,5)); label("A",(-12,0),W); label("B",(0,5),N); label("E",(0,0),SE); [/asy]](http://latex.artofproblemsolving.com/1/3/c/13cae96c6d58989310e8a27b3c7afe7ba5fc6c53.png)
= 
.
= .
Which means 
= 
.
Thus the values of the two diagonals are 
= 
and 
= 
.
Which means area = 
= 
= 
~phoenixfire
See Also
| 2019 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
