2022 AMC 10B Problems/Problem 2
- The following problem is from both the 2022 AMC 10B #2 and 2022 AMC 12B #2, so both problems redirect to this page.
Problem
In rhombus , point
lies on segment
so that
,
, and
. What is the area of
? (Note: The figure is not drawn to scale.)
Solution
(Figure redrawn to scale.)
.
is a rhombus, so
.
is a 3-4-5 right triangle, so
.
The area of the rhombus .
~richiedelgado
Solution 2 (The Area Of A Triangle)
The diagram is the same as solution 1, just constrcuted a line at
When it comes to the sides of a rhombus, their opposite sides are congruent and parallel. Which means
by the Alternate Interior Angles Theorem.
By knowing these statements, we get
.
Knowing that ;
, then we would know that
because
is a 3-4-5 right triangle (as stated in solution 1).
We get the area of as 10 because of the area of the triangle ->
.
Since we know that
and that
. That means we double the area of
,
.
~ghfhgvghj10
See Also
2022 AMC 10B (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 AMC 10 Problems and Solutions |
2022 AMC 12B (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 AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.