Difference between revisions of "FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 2"
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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>. | ||
− | <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== | ||
− | {{AMC12 box|year=2019|ab=B| | + | {{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, is an isosceles right triangle with a right angle at
and with a hypotenuse of
units. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside
.
Solution
The quarter circle is centered at 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
and the midpoint of the hypotenuse, and it is well known that this is half the length of the hypotenuse, or
. The greatest integer less than or equal to
is
, thus we pick answer
.
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 |
The problems on this page are copyrighted by FidgetBoss 4000's Mock American Mathematics Competitions.