2021 AMC 10A Problems/Problem 24
Contents
Problem
The interior of a quadrilateral is bounded by the graphs of and , where a positive real number. What is the area of this region in terms of , valid for all ?
Diagram
Graph in Desmos: https://www.desmos.com/calculator/satawguqsc
~MRENTHUSIASM
Solution 1
The conditions and give and or and . The slopes here are perpendicular, so the quadrilateral is a rectangle. Plug in and graph it. We quickly see that the area is , so the answer can't be or by testing the values they give (test it!). Now plug in . We see using a ruler that the sides of the rectangle are about and . So the area is about . Testing we get which is clearly less than , so it is out. Testing we get which is near our answer, so we leave it. Testing we get , way less than , so it is out. So, the only plausible answer is ~firebolt360
Solution 2 (Casework)
For the equation the cases are
This is a line with -intercept -intercept and slope
This is a line with -intercept -intercept and slope
For the equation the cases are
This is a line with -intercept -intercept and slope
This is a line with -intercept -intercept and slope
Plugging into the choices gives
Plugging into the four above equations and solving systems of equations for intersecting lines [ and and and and ], we get the respective solutions Two solutions follow from here:
Solution 2.1 (Rectangle)
Since the slopes of the intersecting lines (from the four above equations) are negative reciprocals, the quadrilateral is a rectangle. Finally, by the Distance Formula, the length and width of the rectangle are and respectively. The area we seek is
Therefore, the answer is
~MRENTHUSIASM
Solution 2.2 (Shoelace Formula)
Even if we do not recognize that the solutions form the vertices of a rectangle, we can apply the Shoelace Formula to the consecutive vertices from which Therefore, the answer is
Suggested Reading for the Shoelace Formula: https://artofproblemsolving.com/wiki/index.php/Shoelace_Theorem
~MRENTHUSIASM
Solution 3 (Geometry)
Similar to Solution 2, we will use the equations of the four cases:
(1) This is a line with -intercept , -intercept , and slope
(2) This is a line with -intercept , -intercept , and slope
(3)* This is a line with -intercept , -intercept , and slope
(4)* This is a line with -intercept , -intercept , and slope
The area of the rectangle created by the four equations can be written as
=
=
=
(Note: slope )
-fnothing4994
Solution 4 (bruh moment solution)
Trying narrows down the choices to options , and . Trying and eliminates and , to obtain as our answer. -¢
Video Solution by OmegaLearn (System of Equations and Shoelace Formula)
~ pi_is_3.14
See also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.