+159 Given that \(m \angle A= 60^\circ, BC=12\) units, \(\overline{BD} \perp \overline{AC}, \overline{CE} \perp \overline{AB}\) and \(m \angle DBC = 3m \angle ECB\), the length of segment \(EC\) can be expressed in the form \(a(\sqrt{b}+\sqrt{c})\) units where \(b\) and \(c\) have no perfect-square factors. What is the value of \(a+b+c?\)
+2862 Call the center point X.
Quadrilateral AEXD has a total angle sum of 360 degrees.
Since XEA and XDA is 90 degrees, and angle A is 60 degrees.
This means that angle EXD is 90 + 90 + 60 + x = 360
240 + x = 360
x = 120
So angle EXD is 120 degrees.
Based on supplements, EXB is 60 degrees.
Triangles EXB and DXC are 30 60 90 triangles.
Can you find angles XBC and XCB?
Hint: you might need two variables and make a system of equations
+159 I got to a point where I basically knew most of the line segments but I'm trying to upload an image but it doesn't let me.
+2862 Since triangle DBC is a 45 - 45 - 90 triangle.
Can you solve for the length sof BD and DC?
+2862 Notice there are a lot of 90 degree triangles within the diagram,
I think you might have to create a system of equations that can solve for EC using pythagoream theorem knowing the length of DB and DC
+159 
I have finally managed to upload the photo! Here is the stuff that i have solved for.
+159 DB and DC are congruent because of 45-45-90 so their length is \(6\sqrt2\). The image is still under moderation loading at 25% heres the link to Imgur if you don't feel like waiting for the moderation https://imgur.com/a/RLeBs91
+2862 Find AD + DC,
We know the length of BD, so use 30 60 90 to find AD.
then use 30 60 90 triangle to solve for EC,
I think you got it!
+159 AD should just be \(2\sqrt6\) so AD + DC is \(2\sqrt6+6\sqrt2\) but we don't have to do that step because AE is \(4\sqrt6\) since we know that BD is also \(6\sqrt2\) so EC is \(8\sqrt6\) ?
+2862 I dont think AE is congruent to AD
Also, remember your answer is in the form \(a(\sqrt{b}+\sqrt{c})\)
+2862 Welp.
Do you have an AoPS class or something? Because if you submitted that answer then it is big OOOOFFF
I mean, at least you learned something!
+159 Ye I have Aops, but its for alcumus and its apparently a level 24 geometry and I just decided to get stuck on this question. Oh ye, i didnt submit it yet so phew.
+2862 whew if you get that question right you might get like a million XP
+128473
Using the diagram by VooFIX and the Law of Sines we have that
12 / sin (90) = EC / sin (75)
So
12 sin (75) = EC
12 sin (45 + 30) = EC
12 ( sin 45 cos 30 + sin 30 cos 45) = EC
12 ( √2/2 * √3/2 + 1/2 * √2/2)
12 [ √6/4 + √2/4)
(12/4) ( √6 + √2) =
3 (√6 + √2)
So
a + b + c =
3 + 6 + 2 = 11
+2862 just a question, CPhill, was it possible to use 30 60 90 degree triangles to get the same answer?
+159 I have the solution in terms of 30 60 90, but if I post it as a pic its gonna take forever for moderation so you want a link instead?
+2862 you can post the link.
the picture might work faster because CPhill might be here
+128473 Good spot, CU.....yeah....we can do this without Trig
Using VooFIX's diagram.......DC = 6√2 = √72
BD = DC because angles DBC and DCB are equal in right triangle BDC
And since PD is opposite a 30° angle in right triangle PDC....it has 1 / √3 the length of DC =
(1 / √3) * 6 √2 = 2√6 = √24
And
PC^2 = PD^2 + DC^2
PC^2 = 24 + 72
PC^2 = 96
PC = √96 = 4√6
BP = BD - BP = 6√2 - 2√6
And since EP is opposite a 30° angle in right triangle BPE....its length = (1/2) (BP) = 3√2 - √6
So
EC = PC + EP = 4√6 + ( 3√2 - √6) = 3√6 + 3√2 = 3 ( √6 + √2)
+2862 bruhhhhh lol at least my brain got its exercise in for today.
now time to do school homework
I have been postponing it because its way to easy and I feel like a peasant doing slave work when I do my homework.
+159 XD So basically, you gain brain cells when you look at math forums and lose brain cells when you do homework.
+159 I Gotta go too, I have all state auditions in 5 days and Jr. Honors Tom. so practicing while refreshing the page XD
+2862 yup, its a type of survival.
If I don't look at math forums, my IQ will constantly decrease until I am incapable of higher though.
This is to counteract the effects of the bad education system we have today in America.
Ok go practice practice oh life is hard
quote hamlet
to be or not to be
is life worth living?
is it worth the suffering?
we suffer because we are afraid of death.
or https://imgur.com/a/1iQI4o3
