Difference between revisions of "2019 AMC 8 Problems/Problem 4"
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− | Since a rhombus has sides of equal length, <math>AB=BC=CD=DA=\frac{52}{4}=13</math>. In triangle ABC, <math>AB=BC=13</math> and <math>AC=24</math>. Using Heron's formula, we have <math>[ABCD]=[ABC]+[ACD]=2[ABC]=2\sqrt{25 | + | Since a rhombus has sides of equal length, <math>AB=BC=CD=DA=\frac{52}{4}=13</math>. In triangle ABC, <math>AB=BC=13</math> and <math>AC=24</math>. Using Heron's formula, we have <math>[ABCD]=[ABC]+[ACD]=2[ABC]=2\sqrt{25\cdot12\cdot12\cdot1}</math>. Simplifying, we have <math>[ABC]=\sqrt{3600}=60</math> so <math>[ABCD]=2\cdot60=\boxed{\textbf{(D)} 120}</math>. ~~RWhite |
+ | (meadsy69) | ||
==See Also== | ==See Also== |
Revision as of 18:40, 5 July 2020
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 Pythagorean theorem, we find that
=
.
Thus the values of the two diagonals are =
and
=
.
The area of a rhombus is =
=
=
~phoenixfire
Solution 2 (meadsy69)
Since a rhombus has sides of equal length,
. In triangle ABC,
and
. Using Heron's formula, we have
. Simplifying, we have
so
. ~~RWhite
(meadsy69)
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 |
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