2021 AMC 10A Problems/Problem 24
Contents
- 1 Problem
- 2 Diagram
- 3 Solution 1 (Generalized Value of a)
- 4 Solution 2 (Specified Value of a)
- 5 Solution 3 (Slopes and Intercepts)
- 6 Solution 4 (Trigonometry)
- 7 Solution 5 (Observations)
- 8 Solution 6 (Observations)
- 9 Solution 7 (Observations: Cheap)
- 10 Video Solution by OmegaLearn (System of Equations and Shoelace Formula)
- 11 Video Solution by MRENTHUSIASM (English & Chinese)
- 12 See also
Problem
The interior of a quadrilateral is bounded by the graphs of and , where is 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 (Generalized Value of a)
The cases for are or two parallel lines. We rearrange each case and construct the table below: The cases for are or two parallel lines. We rearrange each case and construct the table below: Since the slopes of intersecting lines and are negative reciprocals, we get four right angles, from which the quadrilateral is a rectangle.
Two solutions follow from here:
Solution 1.1 (Distance Between Parallel Lines)
Recall that for constants and the distance between parallel lines is From this formula:
- The distance between lines and is the length of this rectangle.
- The distance between lines and is the width of this rectangle.
The area we seek is ~MRENTHUSIASM
Solution 1.2 (Distance Between Points)
The solutions to systems of equations are respectively, which are the consecutive vertices of this rectangle.
By the Distance Formula, the length and width of this rectangle are and respectively.
The area we seek is ~MRENTHUSIASM
Solution 2 (Specified Value of a)
In this solution, we will refer to equations and from Solution 1.
Substituting into the answer choices gives
At the solutions to systems of equations are respectively, which are the consecutive vertices of the quadrilateral.
Two solutions follow from here:
Solution 2.1 (Area of a Rectangle)
From the tables in Solution 1, we conclude that the quadrilateral is a rectangle.
By the Distance Formula, the length and width of this rectangle are and respectively.
The area we seek is from which the answer is
~MRENTHUSIASM
Solution 2.2 (Area of a General Quadrilateral)
Even if we do not recognize that the quadrilateral is a rectangle, we can apply the Shoelace Theorem to its consecutive vertices The area we seek is from which the answer is
~MRENTHUSIASM
Solution 3 (Slopes and Intercepts)
The quadrilateral is enclosed by four lines. Similar to Solution 1, we will use the equations from the four cases:
- This is a line with -intercept -intercept and slope
- This is a line with -intercept -intercept and slope
- This is a line with -intercept -intercept and slope
- This is a line with -intercept -intercept and slope
It follows that and .
Because the slope of line is , , , .
It follows that and .
Because the slope of line is , , , .
Therefore, the answer is
Solution 4 (Trigonometry)
Similar to Solution 1, we will use the equations from the four cases:
- This is a line with -intercept -intercept and slope
- This is a line with -intercept -intercept and slope
- This is a line with -intercept -intercept and slope
- This is a line with -intercept -intercept and slope
Let The area of the rectangle created by the four equations can be written as ~fnothing4994 (Solution)
~MRENTHUSIASM (Code Adjustments)
Solution 5 (Observations)
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 6 (Observations)
Trying narrows down the choices to options , and . Trying and eliminates and , to obtain as our answer. Refer to Solution 2 for a detailed explanation.
~¢
Solution 7 (Observations: Cheap)
Note that yields different values for all answer choices. If we put in we find that the area of the quadrilateral is This means that the answer must be Refer to Solution 2 for a detailed explanation.
Video Solution by OmegaLearn (System of Equations and Shoelace Formula)
~ pi_is_3.14
Video Solution by MRENTHUSIASM (English & Chinese)
https://www.youtube.com/watch?v=oEY-kX4d87M
~MRENTHUSIASM
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 |
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