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Difference between revisions of "FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 2"

(Created page with "==Problem== <i>(fidgetboss_4000)</i> In the diagram below, <math>ABC</math> is an isosceles right triangle with a right angle at <math>B</math> and with a hypotenuse of <math>...")
 
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<i>(fidgetboss_4000)</i> In the diagram below, <math>ABC</math> is an isosceles right triangle with a right angle at <math>B</math> and with a hypotenuse of <math>40\sqrt2</math>  units. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside <math>\triangle ABC</math>.
 
<i>(fidgetboss_4000)</i> In the diagram below, <math>ABC</math> is an isosceles right triangle with a right angle at <math>B</math> and with a hypotenuse of <math>40\sqrt2</math>  units. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside <math>\triangle ABC</math>.
  
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<math>\textbf{(A)} 26\qquad\textbf{(B)} 27\qquad\textbf{(C)} 28\qquad\textbf{(D)} 29\qquad\textbf{(E)} 30\qquad</math>
+
<math>\textbf{(A) } 26\qquad\textbf{(B) } 27\qquad\textbf{(C) } 28\qquad\textbf{(D) } 29\qquad\textbf{(E) } 30\qquad</math>
  
 
==Solution==
 
==Solution==
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==See also==
 
==See also==
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{{AMC12 box|year=2019|ab=B|num-b=1|num-a=3}}
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{{AMC12 box|year=2019|ab=B|before=[[FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 1]]|num-a=[[FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 3]]}}
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
 
{{FidgetBoss Notice}}
 
{{FidgetBoss Notice}}

Revision as of 15:19, 19 November 2020

Problem

(fidgetboss_4000) In the diagram below, $ABC$ is an isosceles right triangle with a right angle at $B$ and with a hypotenuse of $40\sqrt2$ units. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside $\triangle ABC$.

$\textbf{(A) } 26\qquad\textbf{(B) } 27\qquad\textbf{(C) } 28\qquad\textbf{(D) } 29\qquad\textbf{(E) } 30\qquad$

Solution

The quarter circle is centered at $B$ and just touches the hypotenuse at the midpoint of the hypotenuse due to symmetry in an isosceles right triangle. The value of its radius is equal to the distance between $B$ and the midpoint of the hypotenuse, and it is well known that this is half the length of the hypotenuse, or $\frac{1}{2}\cdot 40\sqrt2=20\sqrt2$. The greatest integer less than or equal to $20\sqrt2$ is $28$, thus we pick answer $\boxed{\textbf{(C) }28}$.

See also

2019 AMC 12B (Problems • Answer Key • Resources)
Preceded by
FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 1 		Followed by
[[2019 AMC 12B Problems/Problem FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 3|Problem FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 3]]
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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