In square ABCD, circle P is tangent to the square and the quarter circ

Question

https://gmatclub.com/forum/download/file.php?id=58297 In square ABCD, shown above, circle P is tangent to the square and the quarter circle centered at A. If CE is tangent to circle P, what is the measure of ∠ECD? A. 10° B. 15° C. 20° D. 25° E. 30° Are You Up For the Challenge: 700 Level Questions Untitled.png

KeyurJoshi · Answer

IMO B But I'm not confident enough on this question, I think its a trap.

Angle CAD + Angle ACD + Angle CDA = 180
45 + 90 + Angle ACD = 180 
Angle ACD = 180-135 = 45

Angle ACD = Angle PCE + Angle ECD 
45 = 30 + Angle ECD
Angle ECD = 15

Pankaj0901 · Answer

How is Angle PCE = 30?

Mona2019 · Answer

I am not able to delete this post. Please refer the post below.

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Mona2019 · Answer

Let length of the side of the square be a
so the radius of the quarter circle is also a
let radius of the smaller circle be r

since OAP=45 degree as it is the angle the diagonal of the square makes with the side,
so  OA=\sqrt{2}r

as  tan45=\frac{OP}{OA} (OP=r)

Let QC be m

so in triangle OCR,  sin OCR=\frac{OR}{OC} =\frac{r}{(r+m)} 
lets find out m in terms of r,
 AO+OQ+QC=AC 
i.e.   \sqrt{2}r +r+m =\sqrt{2}a ----------(1)

AO+OQ =AQ
i.e.  \sqrt{2}r +r = a --------- (2)
 
Substituting equation 2 in 1 we get m=r
so  sin OCR=\frac{r}{2r}=\frac{1}{2} 
OCR=30 degree
and the angle in question is 45-30 =15

LevanKhukhunashvili · Answer

chetan  sir, please spread light on this Buneul-like dark

As GMAT does not test tang-cos there must be another simpler explanation

hadimadi · Answer

Hi,

I found this quite challenging, but maybe it's just me. I was surprised 50% got it correct, or maybe, I just did it too complicated, but here is my way, just using basic knowledge on squares and the pythagorean.

Before, we have to prove co-linearity of lines AC and AP, this can be done and is not too complicated, so just assume it has already been proven and check the following:

In square ABCD, circle P is tangent to the square and the quarter circ

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In square ABCD, circle P is tangent to the square and the quarter circ

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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My Rewards