Difference between revisions of "2024 AMC 12B Problems/Problem 15"
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Thus, the area is:<math>\text{Area} = \frac{1}{2} \left| \log_2 \frac{9}{7} \right|</math> = <math>\boxed{\textbf{(B) }\log_2 \frac{3}{\sqrt{7}}}</math> | Thus, the area is:<math>\text{Area} = \frac{1}{2} \left| \log_2 \frac{9}{7} \right|</math> = <math>\boxed{\textbf{(B) }\log_2 \frac{3}{\sqrt{7}}}</math> | ||
| − | ~Athmyx | + | ~[https://artofproblemsolving.com/wiki/index.php/User:Athmyx Athmyx] |
Revision as of 05:32, 14 November 2024
Problem
A triangle in the coordinate plane has vertices 
, 
, and 
. What is the area of 
?
Solution 1 (Shoelace Theorem)
We rewrite:
.
From here we setup Shoelace Theorem and obtain:
.
Following log properties and simplifying gives (B).
~MendenhallIsBald
Solution 2 (Determinant)
To calculate the area of a triangle formed by three points \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) on a Cartesian coordinate plane, you can use the following formula:
![\[\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|\]](http://latex.artofproblemsolving.com/b/2/5/b25bd42e0a783077028a3464eb3e6dba400bf005.png)
The coordinates are:
, 
, 
Taking a numerical value into account:
![\[\text{Area} = \frac{1}{2} \left| 0 \cdot (2 - 3) + \log_2 3 \cdot (3 - 1) + \log_2 7 \cdot (1 - 2) \right|\]](http://latex.artofproblemsolving.com/7/2/0/72040c1667c3faece7458868ae2345f3acfe74b1.png)
Simplify:
![\[= \frac{1}{2} \left| 0 + \log_2 3 \cdot 2 + \log_2 7 \cdot (-1) \right|\]](http://latex.artofproblemsolving.com/7/e/7/7e77d1541bedfed028dfca6cf59bf1a9cae17cb5.png)
![\[= \frac{1}{2} \left| \log_2 (3^2) - \log_2 7 \right|\]](http://latex.artofproblemsolving.com/e/4/2/e4247b37364ccac9c511eed62d8a1d937661822d.png)
![\[= \frac{1}{2} \left| \log_2 \frac{9}{7} \right|\]](http://latex.artofproblemsolving.com/9/8/8/98862491678c122f9f9ce04df28f9393c0a70ce8.png)
Thus, the area is:
= 
