Difference between revisions of "2021 Fall AMC 10A Problems/Problem 7"
MRENTHUSIASM (talk | contribs) (Created page with "==Problem== As shown in the figure below, point <math>E</math> lies on the opposite half-plane determined by line <math>CD</math> from point <math>A</math> so that <math>\ang...") |
MRENTHUSIASM (talk | contribs) (→Solution) |
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By angle subtraction, we have <math>\angle ADE = 360^\circ - \angle ADC - \angle CDE = 160^\circ.</math> | By angle subtraction, we have <math>\angle ADE = 360^\circ - \angle ADC - \angle CDE = 160^\circ.</math> | ||
| − | Note that <math>\triangle DEF</math> is | + | Note that <math>\triangle DEF</math> is isosceles, so <math>\angle EFD = \frac{180^\circ - \angle ADE}{2}=10^\circ.</math> Finally, we get <math>\angle AFE = 180^\circ - \angle EFD = \boxed{\textbf{(D) }170}</math> degrees. |
~MRENTHUSIASM | ~MRENTHUSIASM | ||
Revision as of 18:52, 22 November 2021
Problem
As shown in the figure below, point 
lies on the opposite half-plane determined by line 
from point 
so that 
. Point 
lies on 
so that 
, and 
is a square. What is the degree measure of 
?
![[asy] usepackage("mathptmx"); size(6cm); pair A = (0,10); label("$A$", A, N); pair B = (0,0); label("$B$", B, S); pair C = (10,0); label("$C$", C, S); pair D = (10,10); label("$D$", D, SW); pair EE = (15,11.8); label("$E$", EE, N); pair F = (3,10); label("$F$", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label("$110^\circ$", (15,9), SW); [/asy]](http://latex.artofproblemsolving.com/1/3/4/134ae4df1e9e5a61162213df0f7339cc588865b9.png)
Solution
By angle subtraction, we have 
Note that 
is isosceles, so 
Finally, we get 
degrees.
~MRENTHUSIASM
See Also
| 2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 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. 
