Difference between revisions of "2019 AMC 8 Problems/Problem 4"
(→Solution 2) |
m (Formatting) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
| − | |||
Quadrilateral <math>ABCD</math> is a rhombus with perimeter <math>52</math> meters. The length of diagonal <math>\overline{AC}</math> is <math>24</math> meters. What is the area in square meters of rhombus <math>ABCD</math>? | Quadrilateral <math>ABCD</math> is a rhombus with perimeter <math>52</math> meters. The length of diagonal <math>\overline{AC}</math> is <math>24</math> meters. What is the area in square meters of rhombus <math>ABCD</math>? | ||
| Line 20: | Line 19: | ||
<math>\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144</math> | <math>\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144</math> | ||
| − | == Solution 1 == | + | ==Solution 1== |
<asy> | <asy> | ||
draw((-12,0)--(0,5)); | draw((-12,0)--(0,5)); | ||
| Line 69: | Line 68: | ||
~math4all | ~math4all | ||
| − | ==Video Solution by Math-X (First fully understand the problem | + | ==Video Solutions== |
| + | |||
| + | ===Video Solution 1 (Detailed Explanation)=== | ||
| + | https://youtu.be/nbcPvG024_o | ||
| + | |||
| + | ~ChillGuyDoesMath | ||
| + | |||
| + | ===Video Solution 2 by Math-X (First fully understand the problem)=== | ||
https://youtu.be/IgpayYB48C4?si=CGr1gRwXKdjrhl24&t=678 | https://youtu.be/IgpayYB48C4?si=CGr1gRwXKdjrhl24&t=678 | ||
~Math-X | ~Math-X | ||
| − | == Video Solution == | + | ===Video Solution 3=== |
| − | |||
The Learning Royal: https://youtu.be/IiFFDDITE6Q | The Learning Royal: https://youtu.be/IiFFDDITE6Q | ||
| − | == Video Solution | + | ===Video Solution 4=== |
| − | + | https://www.youtube.com/watch?v=-yHfOUapg7I&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=5 | |
| − | |||
| − | ==Video Solution | + | ===Video Solution 5=== |
https://youtu.be/mL6gIb5y3B0 | https://youtu.be/mL6gIb5y3B0 | ||
~savannahsolver | ~savannahsolver | ||
| − | ==Video Solution (CREATIVE THINKING | + | ===Video Solution 6 (CREATIVE THINKING)=== |
https://youtu.be/UCaEXbe7mN0 | https://youtu.be/UCaEXbe7mN0 | ||
~Education, the Study of Everything | ~Education, the Study of Everything | ||
| − | ==Video Solution by The Power of Logic(1 to 25 Full Solution)== | + | ===Video Solution 7 by The Power of Logic(1 to 25 Full Solution)=== |
https://youtu.be/Xm4ZGND9WoY | https://youtu.be/Xm4ZGND9WoY | ||
Latest revision as of 14:48, 1 January 2025
Problem
Quadrilateral 
is a rhombus with perimeter 
meters. The length of diagonal 
is 
meters. What is the area in square meters of rhombus ?
![[asy] draw((-13,0)--(0,5)); draw((0,5)--(13,0)); draw((13,0)--(0,-5)); draw((0,-5)--(-13,0)); dot((-13,0)); dot((0,5)); dot((13,0)); dot((0,-5)); label("A",(-13,0),W); label("B",(0,5),N); label("C",(13,0),E); label("D",(0,-5),S); [/asy]](http://latex.artofproblemsolving.com/e/5/4/e54b35c6e9f6a1eaa16a3138ed06ebc73122be63.png)
Solution 1
![[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)
A rhombus has sides of equal length. Because the perimeter of the rhombus is , each side is 
. In a rhombus, diagonals are perpendicular and bisect each other, which means 
= 
= 
.
Consider one of the
![[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)
= 
, and = . Using the Pythagorean theorem, we find that 
= 
.
You know the Pythagorean triple, (5, 12, 13).
Thus the values of the two diagonals are = and 
= 
.
The area of a rhombus is = 
= 
= 
Solution 2
Right off the bat, we can see that the perimeter of the figure is 52. Dividing this by four, we can get that each side is equal to 13. By drawing a line perpendicular to the one given, we can split the figure into four right triangles. 12 (24/2) is equal to the height of one small right triangle, and 13 is the slanted side. Using the Pythagorean theorem we can find that 169 (13 squared) - 144 (12 squared) = 25 (five squared). With this, we can determine that each small right triangle equals 30. Multiplying that by four we can get
~math4all
Video Solutions
Video Solution 1 (Detailed Explanation)
~ChillGuyDoesMath
Video Solution 2 by Math-X (First fully understand the problem)
https://youtu.be/IgpayYB48C4?si=CGr1gRwXKdjrhl24&t=678
~Math-X
Video Solution 3
The Learning Royal: https://youtu.be/IiFFDDITE6Q
Video Solution 4
https://www.youtube.com/watch?v=-yHfOUapg7I&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=5
Video Solution 5
~savannahsolver
Video Solution 6 (CREATIVE THINKING)
~Education, the Study of Everything
Video Solution 7 by The Power of Logic(1 to 25 Full Solution)
~Hayabusa1
See also
| 2019 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 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. 
