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655-705 (Hard)|   Geometry|      
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Bunuel
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In the diagram below, ABCD is a square with a circle inscribed inside. 17 Jul 2017, 23:36
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65% (hard)

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55% (01:55) correct 45% (02:09) wrong based on 135 sessions

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In the diagram below, ABCD is a square with a circle inscribed inside. P is a point on the circle. A rectangle AXPY is constructed with A and P as the opposite vertices of the rectangle. What is the length of a side of the square?



(1) AP = 2√2
(2) The line AP coincides with the diagonal AC.

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2017-07-18_1034_001.png
2017-07-18_1034_001.png [ 7.57 KiB | Viewed 3755 times ]
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C
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In the diagram below, ABCD is a square with a circle inscribed inside. 18 Jul 2017, 02:50
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Option 1. is not helpful as point P can be anywhere in the circle
Option 2. For AP to coincide with AC the rectangle AXPY has to be a square. But this too doesnt help us find the lenght of side of the square

Diameter of the circle=side of square=a
Combined, AP=2√2, AC=2√2+a+2√2=a+4√2
Diagonal of a square is a√2, therefore a√2=a+4√2
a=(4√2)/(√2-1)

C IMO

Bunuel
In the diagram below, ABCD is a square with a circle inscribed inside. P is a point on the circle. A rectangle AXPY is constructed with A and P as the opposite vertices of the rectangle. What is the length of a side of the square?



(1) AP = 2√2
(2) The line AP coincides with the diagonal AC.

Show Spoiler
Attachment:
2017-07-18_1034_001.png
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In the diagram below, ABCD is a square with a circle inscribed inside. 01 Jul 2019, 15:23
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origen87
Option 1. is not helpful as point P can be anywhere in the circle
Option 2. For AP to coincide with AC the rectangle AXPY has to be a square. But this too doesnt help us find the lenght of side of the square

Diameter of the circle=side of square=a
Combined, AP=2√2, AC=2√2+a+2√2=a+4√2
Diagonal of a square is a√2, therefore a√2=a+4√2
a=(4√2)/(√2-1)

C IMO

Bunuel
In the diagram below, ABCD is a square with a circle inscribed inside. P is a point on the circle. A rectangle AXPY is constructed with A and P as the opposite vertices of the rectangle. What is the length of a side of the square?



(1) AP = 2√2
(2) The line AP coincides with the diagonal AC.

Show Spoiler
Attachment:
2017-07-18_1034_001.png
Show more


Hi Bunuel & origen87, I know the post is a little old but could you explain the highlighted portion please, "Diagonal of a square is a√2". Is this some sort of rule I am not aware of?

Thanks in advance!
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In the diagram below, ABCD is a square with a circle inscribed inside. 01 Jul 2019, 18:54
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aliakberza

Hi there, not who you asked but to answer your question, there is the rule that the diagonal of a square with side length a is a√2. This is the same rule that applies to an isosoles right triangle with side lengths of a, a, and a√2.

This solution defines the diagonal in two ways: it says AC is diagonal which equals a+4√2 and it also says diagonal can be calculated as a√2. Two equations, one unknown, we can solve for the unknown.

Hope it helps!

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In the diagram below, ABCD is a square with a circle inscribed inside. 02 Jul 2019, 04:55
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fizzypopcorn
aliakberza

Hi there, not who you asked but to answer your question, there is the rule that the diagonal of a square with side length a is a√2. This is the same rule that applies to an isosoles right triangle with side lengths of a, a, and a√2.

This solution defines the diagonal in two ways: it says AC is diagonal which equals a+4√2 and it also says diagonal can be calculated as a√2. Two equations, one unknown, we can solve for the unknown.

Hope it helps!

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Hey thanks! Figured it out after reading your post and going over the solution again. I had got the side of the smaller square using the same rule but somehow could not make the connection that the same would apply to the big square too. The diameter equals side of the larger square is what I was missing. Appreciate your help. :)
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In the diagram below, ABCD is a square with a circle inscribed inside. 21 Dec 2023, 23:21
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