Difference between revisions of "2019 AMC 8 Problems/Problem 4"
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− | <math>\overline{AB}</math> = <math>13</math>, and <math>\overline{AE}</math> = <math>12</math>. Using Pythagorean theorem, we find that <math>\overline{BE}</math> = <math>5</math>. | + | <math>\overline{AB}</math> = <math>13</math>, and <math>\overline{AE}</math> = <math>12</math>. Using the Pythagorean theorem, we find that <math>\overline{BE}</math> = <math>5</math>. |
− | You | + | You know the Pythagorean triple, (5, 12, 13). |
Thus the values of the two diagonals are <math>\overline{AC}</math> = <math>24</math> and <math>\overline{BD}</math> = <math>10</math>. | Thus the values of the two diagonals are <math>\overline{AC}</math> = <math>24</math> and <math>\overline{BD}</math> = <math>10</math>. |
Revision as of 17:40, 1 January 2022
Problem 4
Quadrilateral is a rhombus with perimeter
meters. The length of diagonal
is
meters. What is the area in square meters of rhombus
?
Solution 1
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 right triangles:
=
, 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 =
=
=
Video Solution
The Learning Royal: https://youtu.be/IiFFDDITE6Q
Video Solution 2
Solution detailing how to solve the problem:https://www.youtube.com/watch?v=-yHfOUapg7I&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=5
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 |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.