Difference between revisions of "2024 AMC 12B Problems/Problem 15"
(→Solution 1 (Shoelace Theorem)) |
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| − | ==Solution 2== | + | ==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: | 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: | ||
| − | + | <cmath> | |
\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| | \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| | ||
| − | + | </cmath> | |
The coordinates are: | The coordinates are: | ||
| − | - | + | - <cmath>A(0, 1)</cmath> |
| − | - | + | - <cmath>B(\log_2 3, 2)</cmath> |
| − | - | + | - <cmath>C(\log_2 7, 3)</cmath> |
| − | + | Taking a numerical value into account: | |
| − | + | <cmath> | |
\text{Area} = \frac{1}{2} \left| 0 \cdot (2 - 3) + \log_2 3 \cdot (3 - 1) + \log_2 7 \cdot (1 - 2) \right| | \text{Area} = \frac{1}{2} \left| 0 \cdot (2 - 3) + \log_2 3 \cdot (3 - 1) + \log_2 7 \cdot (1 - 2) \right| | ||
| − | + | </cmath> | |
Simplify: | Simplify: | ||
| − | + | <cmath> | |
= \frac{1}{2} \left| 0 + \log_2 3 \cdot 2 + \log_2 7 \cdot (-1) \right| | = \frac{1}{2} \left| 0 + \log_2 3 \cdot 2 + \log_2 7 \cdot (-1) \right| | ||
| − | + | </cmath> | |
| − | + | <cmath> | |
= \frac{1}{2} \left| \log_2 (3^2) - \log_2 7 \right| | = \frac{1}{2} \left| \log_2 (3^2) - \log_2 7 \right| | ||
| − | + | </cmath> | |
| − | + | <cmath> | |
= \frac{1}{2} \left| \log_2 \frac{9}{7} \right| | = \frac{1}{2} \left| \log_2 \frac{9}{7} \right| | ||
| − | + | </cmath> | |
Thus, the exact area is: | Thus, the exact area is: | ||
| − | + | <cmath> | |
\text{Area} = \frac{1}{2} \left| \log_2 \frac{9}{7} \right| | \text{Area} = \frac{1}{2} \left| \log_2 \frac{9}{7} \right| | ||
| − | + | </cmath> | |
| − | + | <cmath> | |
\boxed{\text{B. \log_2 \frac{3}{\sqrt{7}}}} | \boxed{\text{B. \log_2 \frac{3}{\sqrt{7}}}} | ||
| − | + | </cmath> | |
| + | |||
| + | --- | ||
| + | |||
| + | This format with `<math></math>` will make Obsidian recognize all the mathematical expressions as standalone blocks, rendering them properly in mathematical form. | ||
Revision as of 05:23, 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:
- ![\[A(0, 1)\]](http://latex.artofproblemsolving.com/2/9/7/297759e2b514ca40da44cf1a0bc93418ac760c10.png)
- ![\[B(\log_2 3, 2)\]](http://latex.artofproblemsolving.com/2/4/2/242b53c54fc41068d7bc87fdb19d6aefda2ea7c8.png)
- ![\[C(\log_2 7, 3)\]](http://latex.artofproblemsolving.com/0/e/5/0e5c4e12a54fc547b5d125e1dc13ba41280f411a.png)
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 exact area is:
![\[\text{Area} = \frac{1}{2} \left| \log_2 \frac{9}{7} \right|\]](http://latex.artofproblemsolving.com/0/c/b/0cbea38171ce0c749ea849ae6776f94fe195be0c.png)
\[\boxed{\text{B. \log_2 \frac{3}{\sqrt{7}}}}\] (Error compiling LaTeX. Unknown error_msg)
---
This format with `$$ (Error compiling LaTeX. Unknown error_msg)` will make Obsidian recognize all the mathematical expressions as standalone blocks, rendering them properly in mathematical form.
