1979 AHSME Problems/Problem 3: Difference between revisions

Latest revision as of 23:42, 31 December 2022

Problem 3

$[asy] real s=sqrt(3)/2; draw(box((0,0),(1,1))); draw((1+s,0.5)--(1,1)); draw((1+s,0.5)--(1,0)); draw((0,1)--(1+s,0.5)); label("$A$",(1,1),N); label("$B$",(1,0),S); label("$C$",(0,0),W); label("$D$",(0,1),W); label("$E$",(1+s,0.5),E); //Credit to TheMaskedMagician for the diagram[/asy]$

In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$ . What is the measure of $\measuredangle AED$ in degrees?

$\textbf{(A) }10\qquad \textbf{(B) }12.5\qquad \textbf{(C) }15\qquad \textbf{(D) }20\qquad \textbf{(E) }25$

Solution

Solution by e_power_pi_times_i

Notice that $\measuredangle DAE = 90^\circ+60^\circ = 150^\circ$ and that $AD = AE$ . Then triangle $ADE$ is isosceles, so $\measuredangle AED = \dfrac{180^\circ-150^\circ}{2} = \boxed{\textbf{(C) } 15}$ .

Solution 2 (Obnoxiously tedious)

WLOG, let the side length of the square and the equilateral triangle be $1$ . $\angle{DAE}=90^\circ+60^\circ=150^\circ$ . Apply the law of cosines then the law of sines, we find that $\angle{AED}=15^\circ$ . Select $\boxed{C}$ .

Please don't try this solution. This is just obnoxiously tedious and impractical.

~hastapasta

Video Solution by OmegaLearn

https://youtu.be/FDgcLW4frg8?t=1341

~ pi_is_3.14

1979 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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All AHSME Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

@@ Line 27: / Line 27: @@
 Notice that <math>\measuredangle DAE = 90^\circ+60^\circ = 150^\circ</math> and that <math>AD = AE</math>. Then triangle <math>ADE</math> is isosceles, so <math>\measuredangle AED = \dfrac{180^\circ-150^\circ}{2} = \boxed{\textbf{(C) } 15}</math>.
+== Solution 2 (Obnoxiously tedious) ==
+WLOG, let the side length of the square and the equilateral triangle be <math>1</math>. <math>\angle{DAE}=90^\circ+60^\circ=150^\circ</math>. Apply the law of cosines then the law of sines, we find that <math>\angle{AED}=15^\circ</math>. Select <math>\boxed{C}</math>.
+Please don't try this solution. This is just obnoxiously tedious and impractical.
+~hastapasta
+== Video Solution by OmegaLearn==
+https://youtu.be/FDgcLW4frg8?t=1341
+~ pi_is_3.14
 == See also ==
-{{AHSME box|year=1979|before=num-b=1|num-a=4}}
+{{AHSME box|year=1979|num-b=2|num-a=4}}
-[[Category:Introductory Algebra Problems]]
+[[Category:Introductory Geometry Problems]]
 {{MAA Notice}}

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