The circle above circumscribes square ABCD. What is the valu

Question

T5316.png The circle above circumscribes square ABCD. What is the value of w°+x°+y°+z°? A. 180° B. 135° C. 120° D. 100° E. 90°

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=12641 The circle above circumscribes square ABCD. What is the value of w°+x°+y°+z°? A. 180° B. 135° C. 120° D. 100° E. 90° The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. So the central angle which subtends the minor arc BC equals to twice the angle x+z (as inscribed angle x plus inscribed angle z subtend the same minor arc BC). Now, minor arc BC is 1/4 of the circumference hence the central angle which subtends it equals to 360/4=90 --> 90=2(x+z) --> x+z=45; Similarly 90=2(w+y) (for minor arc AD) --> w+y=45; x+z+w+y=90. Answer: E. For more on circles and angles check: math-circles-87957.html Hope it helps.

hemanthp · Answer

E it is. 
how:
We all agree that X+Z = W+Y (draw parallel lines at X and Y and you see that the alternate angles match up giving you this condition)

From the figure you can see that the angles subtended by at the center by those arcs that subtend X and Z at the circumference will be 2X and 2Z. Which is the same angle subtended by the chord BC at the center which is 90 for a square.

Similarly for W+Y.

so 2(X+Z) = 90 or X+Z = 45. And therefore X+Z+W+Y = 90.

Clear as spring water. Good question.
---
+1 Kudos if you like this!

ezhilkumarank · Answer

Wow. Really good one :-D

-- finding it hard to come up some sort of a approach to this problem. Just curious -- what is the source of this question.

rafi · Answer

It's from a Master GMAT test

babyif19 · Answer

This is quite a hard question -

I tried to set up a few equations and substitute angles out to hope to get the total value but I onl got x-w-y+z=0 which means x+y+w+z=2(x+z) =2(w+y).

If you imagine the center of the square, called it O. and imagine where Y hits the circle, call it E. BOE=2X and BOC=2Z (Angles that share the same arc are of same value and one with the center of cicle as the top point is always twice as the one with an arbitraty point on the circle) and BOC=90 from the square therefore, w+x+y+z is 90.

It took me more than 3 minutes to solve this and definitely is not recommended on the test. I would have just guessed something is a divident or multiples of 90, given the circle and the square. A, C or E are in the run. If you also look at the image, I wouls have guessed either 90 or 120. The rest of it is your luck under pressure.

Hope this helps.

rafi · Answer

Thanks for the answer. But I still don't get it. BOC is not on the same arc as angle Z. If I used your E (where y hits the circle) I'd say EOC is on the same arc. Anyhow I still don't understand how you went from this data to the conclusion that the 4 angles equals 90 degrees.
Thanks again!

hemanthp · Answer

Babyif, I think you are naming the variables incorrectly. As Rafi pointed out...

rafi · Answer

Excellent answer! Kudos!

babyif19 · Answer

Hi guys:
I might have named the angles wrong but on the way to work this morning - I thought about quick way to solve this problem without drawing parallel lines or set up equations. Using algebra to solve geometry problems usually takes longer, nevertheless should arrive the same conclusion.

Ready? I'd love to draw it out as Heman did but I have no scanner in hand now.

I am going to call the left unnamed point in on the circle E ( between A and D). And similarly the right one F. Center of the circle which is also center of the square is O. So connect OA, OE and OD. You will get three lines similar to what Rafi drew. BOF=2x,COF=2z, AOE=2W and EOD=2Y. Because BOF and X share the same length of arc and similarly the other three pairs. It's easy to know that BOF+COF=BOC=90 from the property of square ( or circle). And AOE+EOD=AOD=90; 2x+2Z+2W+2Y= 90+90=180. Thus answer is 90.

This should be how the test writer wanted us to solve the problem. Other than lucky guesses, this is the fastest way I can think of. And it did not come to me immediately. Hope this explanation is easier to read. Very hard to explain a geormetry problem by words only. hehe.

Posted from my mobile device

hemanthp · Answer

I think Babyif got it right the first time. And I didn't use a scanner dude. It is PAINT

.

wvivek · Answer

Hii,

I think there is a simpler way to get this,

See ABCD is a square so we immediately know all the four arcs are same adn corresponding angles formed by square are 90 degree Now if we assume a diagonal BD the angle BDC is 45 ( In square diagonal bisects opposite angles) and this angle subtends the same arc as X & Z since same arc is subtended the sum X + Z = 45

The same logic works for the other two as well and the sum is 90.

Vivek

tingle15 · Answer

@vivek: Very good use of concepts...

hirendhanak · Answer

@Bunuel, Can you help with this

saurabhgoel · Answer

Hi,
May be it sounds funny, but by observing the diagram i found it out that 90degree angle is broken into different acute angles w, x, y, z. So if we observe angle ABC = 90, it is broken into w, x, y, z in the order. And i got the correct OA.

Now by seeing so much explanations from everyone, i am confused whether my first observation was correct or not ?

Plz help.

madn800 · Answer

Hemmanthp explained the solution well but couldn't explain as to why 2(x+z)=90. Bunuel did it well to explain it.

vipulgoel · Answer

Hi Bunuel,

In analogy to central angle theorem will it be true??

Please refer attachment

pulkitaggi · Answer

I guess there is an easier way to do this. 
Since the points where angle X and Y are made are arbitrarily chosen, and since there can only be a single answer to this question ( Problem solving), Move the point X ( point where angle x is made) as close to point D as possible- leave only infinitesimally small space between point X and D. 
If you Redraw the image, you can see that Y almost coincides with C. From this -

W approximately = 45
X approximately = 45 
Z approximately = 0 (as Line DY moves towards DC, the angle Z collapses to zero) 
Y approximately = 0 ( As line XY moves towards DC, angle Y collapses to zero)

PS- point X= the point where angle x is formed, and point Y = the point where angle y is formed

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The circle above circumscribes square ABCD. What is the valu

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