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.
Contents
Problem
In rhombus , point lies on segment so that , , and . What is the area of ? (Note: The figure is not drawn to scale.)
Solution 1
.
is a rhombus, so .
is a right triangle, hence .
The area of the rhombus is base times height: .
~richiedelgado
Solution 2 (The Area Of A Triangle)
The diagram is from as Solution 1, but a line is constructed at .
When it comes to the sides of a rhombus, their opposite sides are congruent and parallel. This means that , by the Alternate Interior Angles Theorem.
By SAS Congruence, we get .
Since and , we know that because is a 3-4-5 right triangle, as stated in Solution 1.
The area of would be , since the area of the triangle is .
Since we know that and that , so we can double the area of to get .
~ghfhgvghj10, minor edits by MinecraftPlayer404
Video Solution 1
~Education, the Study of Everything
Video Solution(1-16)
~~Hayabusa1
Video Solution by Interstigation
https://youtu.be/_KNR0JV5rdI?t=97
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.