Difference between revisions of "2018 AMC 12B Problems/Problem 21"
(Created page with "== Problem == In <math>\triangle{ABC}</math> with side lengths <math>AB = 13</math>, <math>AC = 12</math>, and <math>BC = 5</math>, let <math>O</math> and <math>I</math> deno...") |
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<cmath>x^2=(5+12-13)x</cmath><cmath>x\neq 0\implies x=4.</cmath>Now the area of our triangle can be calculated with the Shoelace Theorem. The answer turns out to be <math>\boxed{\textbf{E}.}</math> | <cmath>x^2=(5+12-13)x</cmath><cmath>x\neq 0\implies x=4.</cmath>Now the area of our triangle can be calculated with the Shoelace Theorem. The answer turns out to be <math>\boxed{\textbf{E}.}</math> | ||
+ | == Solution == | ||
+ | |||
+ | Notice that we can let <math>M = C</math>. If <math>M = \left(0, 0\right)</math>, then <math>C = \left(6, -\frac{5}{2}\right)</math> and <math>I = \left(4, -\frac{1}{2}\right)</math>. Using shoelace formula, we get <math>\left[COI\right] = \frac{7}{2}</math>. <math>\boxed{\textbf E.}</math> | ||
==See Also== | ==See Also== |
Revision as of 16:25, 16 February 2018
Contents
Problem
In with side lengths , , and , let and denote the circumcenter and incenter, respectively. A circle with center is tangent to the legs and and to the circumcircle of . What is the area of ?
Solution
Let the triangle have coordinates Then the coordinates of the incenter and circumcenter are and respectively. If we let then satisfies Now the area of our triangle can be calculated with the Shoelace Theorem. The answer turns out to be
Solution
Notice that we can let . If , then and . Using shoelace formula, we get .
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 12 Problems and Solutions |
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