Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2019 AMC 10A Problems/Problem 13"

(→‎Solution)
(→‎Solution)
Line 6: Line 6:
  
 
==Solution==
 
==Solution==
−
[asy]unitsize(40);draw((-1,0)--(1,0)--(0,2.75)--cycle);draw(circumcircle((-1,0),(0,0),(0,2.75)));label("<math>A</math>",(1,0),SE);label("<math>C</math>",(0,2.75),N);label("<math>B</math>",(-1,0),SW);label("<math>E</math>",(0,0),S);label("<math>D</math>",(0.77,0.64),E);draw((0,0)--(0,2.75));draw((-1,0)--(0.77,0.64));[/asy]
 
 
Drawing it out, we see <math>\angle BDC</math> and <math>\angle BEC</math> are right angles, as they are inscribed in a semicircle. Noting that they subtend to the same arc of <math>180^{\circ}</math>, we can find <math>\angle ECB=20</math> and <math>\angle DBC=50</math> by the triangle angle sum. Then we take triangle <math>BFC</math>, and find <math>\angle BFC=180-50-20=\boxed{(\text{D})110}.</math>
 
Drawing it out, we see <math>\angle BDC</math> and <math>\angle BEC</math> are right angles, as they are inscribed in a semicircle. Noting that they subtend to the same arc of <math>180^{\circ}</math>, we can find <math>\angle ECB=20</math> and <math>\angle DBC=50</math> by the triangle angle sum. Then we take triangle <math>BFC</math>, and find <math>\angle BFC=180-50-20=\boxed{(\text{D})110}.</math>
  

Revision as of 19:42, 9 February 2019

Problem

Let $\Delta ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$. Contruct the circle with diameter $\overline{BC}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$. What is the degree measure of $\angle BFC ?$

$\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$

Solution

Drawing it out, we see $\angle BDC$ and $\angle BEC$ are right angles, as they are inscribed in a semicircle. Noting that they subtend to the same arc of $180^{\circ}$, we can find $\angle ECB=20$ and $\angle DBC=50$ by the triangle angle sum. Then we take triangle $BFC$, and find $\angle BFC=180-50-20=\boxed{(\text{D})110}.$

See Also

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10A_Problems/Problem_13&oldid=101607"