Difference between revisions of "2024 AMC 12B Problems/Problem 15"
(Created page with "==Solution (Shoelace Theorem)== We rewrite: <math>A(0,1)</math> <math>B(\log _{2} 3, 2)</math> <math>C(\log _{2} 7, 3)</math> From here we setup Shoelace Theorem and obtain: <...") |
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| − | ==Solution (Shoelace Theorem)== | + | ==Problem== |
| + | A triangle in the coordinate plane has vertices <math>A(\log_21,\log_22)</math>, <math>B(\log_23,\log_24)</math>, and <math>C(\log_27,\log_28)</math>. What is the area of <math>\triangle ABC</math>? | ||
| + | |||
| + | <math> | ||
| + | \textbf{(A) }\log_2\frac{\sqrt3}7\qquad | ||
| + | \textbf{(B) }\log_2\frac3{\sqrt7}\qquad | ||
| + | \textbf{(C) }\log_2\frac7{\sqrt3}\qquad | ||
| + | \textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad | ||
| + | \textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad | ||
| + | </math> | ||
| + | |||
| + | |||
| + | ==Solution 1 (Shoelace Theorem)== | ||
We rewrite: | We rewrite: | ||
<math>A(0,1)</math> | <math>A(0,1)</math> | ||
<math>B(\log _{2} 3, 2)</math> | <math>B(\log _{2} 3, 2)</math> | ||
| − | <math>C(\log _{2} 7, 3)</math> | + | <math>C(\log _{2} 7, 3)</math>. |
| + | |||
From here we setup Shoelace Theorem and obtain: | From here we setup Shoelace Theorem and obtain: | ||
| − | <math>\frac{1}{2}(2(\log _{2} 3) - log _{2} 7)</math> | + | <math>\frac{1}{2}(2(\log _{2} 3) - log _{2} 7)</math>. |
| + | |||
Following log properties and simplifying gives (B). | Following log properties and simplifying gives (B). | ||
| − | ~ | + | |
| + | ~MendenhallIsBald | ||
Revision as of 01:06, 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
