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HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a
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Bunuel
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HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:00
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Two quarter circles are drawn in a square and a circle is inscribed between the quarter circles. If the side of the square is 2 cm, what is the area of the circle?


A. \(\frac{1}{9}*\pi\)

B. \(\frac{9}{64}*\pi\)

C. \(\frac{3}{8}*\pi\)

D. \(\frac{9}{16}*\pi\)

E. \(\frac{3}{4}*\pi\)


 

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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:36
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ON= OM=r, AB=2

AM = 1 (due to symmetry)

By inspection, it can be observed that the circle with center O touches the circle with center A, internally at N. If two circles touch each other internally, then the distance between their centers is the difference of their radii.

AN=AB= 2

OA = Distance between the centers of circles = AN-ON= 2-r

From the right angled triangle AOM,

OA²= AM²+OM²

(2-r)² = (1)²+r²

4+r²-4r = 1+r²

1-r = 1/4

r = 3/4

Area of circle= \(pi*r^2\) = \(pi*\frac{9}{16}\) OPTION D
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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:17
Given:Two quarter circles are drawn in a square and a circle is inscribed between the quarter circles.
Asked:If the side of the square is 2 cm, what is the area of the circle?

Area of the circle is more than half that of area of the square but less than area of the square
2<Area of circle<4

A. Pi/9 = .35
Incorrect

B. 9pi/64=.44
Incorrect

C. 3pi/8=1.17
Incorrect

D. 9pi/16=1.77
Incorrect

E. 3pi/4 = 2.35
Correct

IMO E

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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:23
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Draw 2 tangent to the inscribed circle at a point where quarter circle touches it. And then draw a perpendicular to those 2 tangents. These perpendicular lines will pass through the center of the circle. So, the point where these 2 perpendicular lines will intersect will be the center of circle.

Let r be the radius of inscribed circle.

So, Applying Pythagoras Theorem, (Refer to image below)

(2-r)^2 = ( 1^2 ) + r^2

Solving we get, r = 3/4

Therefore, Area = π(9/16)

Answer is Option D

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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:24
Correct answer is D.

As radius of highlighted circle will be less than 1.
It will be 3/4 of half of side of square.
Hence area of circle will be π*(3/4)^2=9π/16.

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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:26
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Explanation:

O is the centre of inscribed circle.
Consider one Quarter circle, perpendicular to this tangent will pass through the centre of Quarter circle to Point A.
For other Quarter circle, we do similar construction, the perpendicular to this tangent will pass through the centre of Quarter circle to Point B.
Intersection of these line must contains center O.

As sqauare side is 2, means radius of Quarter circle is 2.

Let inscribed cicle radius is r. So distance between 2 centre is 1-r.

Draw a perpendicular from radius of inscribed circle. Which will bisect side of Square at point E.

Using Pythagoras theorem on Triangle AOE.

1^2 + r^2 = (2-r)^2
4r=4-1
r=3/4

Area of circle: πr^2
= π(3/4)^2
= 9π/16

IMO-D

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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:26
Area of one quarter circle = 4pi/4 =pi= approximately 3

Area of two quarter- circle = 2pi
The shaded circle is almost half of the two quarter-circles, approximately 3.


A. 3/9.....too small

B. 27/64...small

C. 9/8...the circle is between 2-3.

D.27/16...a lil bit more

E. 9/4..yes that's a lil more than 2 and could be the answer

E is the answer.

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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:27
E

Area has to be around half of square as per estimate from scale.

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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:28
D seems to be the correct answer option.

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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:28
Answer:D
I just divided the whole square into four parts for approx calculation of the area of circle so i got dismeter as 1.5 or 3/4
πr^2 i.e π9/16

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HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  Updated on: 05 Sep 2020, 13:06
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unlike the similar one posted recently, this can be answered in under 2 mins :)


Answer : D

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Originally posted by petrichor on 03 Sep 2020, 19:30.
Last edited by petrichor on 05 Sep 2020, 13:06, edited 3 times in total.
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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:31
Two quarter circles are drawn in a square and a circle is inscribed between the quarter circles. If the side of the square is 2 cm, what is the area of the circle?


A. 19∗π19∗π

B. 964∗π964∗π...ans

C. 38∗π38∗π

D. 916∗π916∗π

E. 34∗π
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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:32
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IMO D

Refer attached diagram when we draw two tangents on highlighted circle,

Radius of Quadrant will pass through the center of Circle as referred ; as intersection of two line perpendicular line on tangent passes through center of circle

In Triangle OGA

OG=r (required radius)

OA=2-r

AG=1 (Half of side, Symmetry can also be seen)

SO applying Pythagoras theorem

=>4+r^2-4r =r^2 +1
=>r=3/4
Area =PI*r^2

=pi*9/16



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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:33
First of all the key to solve this question is to find the radius of the circle.

Solving for radius of the circle and from Pythagoras Theorem we get

(2-r)^2 = r^2 + 1
=> r = 3/4

Area of the circle = Pi * (3/4)^2
= 9/16 pi

Hence, D is the answer.
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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:37
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D is the answer please check attached image
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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:42
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First draw a line from corner of the square to circle.

Pythagoras theorem:

\((2-r)^2=1^2+r^2\)------r=\(\frac{3}{4}\)

Area of circle=π*(9/16)

Answer D

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HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  Updated on: 04 Sep 2020, 20:04
We can not calculate r with angle method because we don’t have length of MR

But we have 90 degrees to base and can use Pythagoras theorem with AR

So apply Pythogoras theorem:
O is centre of circle and
point A to end of tangent on right side would be =2 ( radius of quarter circle= side of square)

(AR)^2+r^2= (OA)^2
1+r^2= (2-r)^2
==> 1+r2= 4+r2-4r
r= 3/4

Area of circle = pie*9/16

hence D
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Originally posted by mSKR on 03 Sep 2020, 19:43.
Last edited by mSKR on 04 Sep 2020, 20:04, edited 1 time in total.
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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:43
Used estimation.
Since the side of square is 2cm, diameter of the Circle is approximately 1.5cm.
Radius = 0.75 = 3/4
Area = (9/16)π

IMO D

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Re: HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:50
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The answer is D

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HOT Competition 3 Sep/8PM: Two quarter circles are drawn in a square a [#permalink] New post  03 Sep 2020, 19:51
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Both the lines joining the corners will meet the circle of square as circle inscribed in quarter circle will have tangent at 90
Thus from this
Taking radius of inner circle as r
Right angle triangle which bisect the side is formed
(2-r)^2 = 1+r^2

4+r2-4r = 1+r^2

r = 3/4
Area pi *9/16
IMO D

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