How do you inscribe a semicircle into a square? And the semicircle's diameter can not be the one of the diagonals of the square. It has to look like the picture below.

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- February 21st 2010, 01:19 PMarrowhead566Inscribe a semicircle into a square
How do you inscribe a semicircle into a square? And the semicircle's diameter can not be the one of the diagonals of the square. It has to look like the picture below.

- February 21st 2010, 03:53 PMDiagonal
Very messy unless someone else has a solution. I don't even really count this as a solution. However, if you use the following diagram, then from the triangle inside the lower right sector, you can find x in terns of r. Then you have sides of a right triangle that will relate r and d. That is, you can form the ratio of r and d. However, I did a little of the math, and it is VERY messy [I might easily have missed something.] Further, that does not at all imply a drafting solution, but it's the best I can come up with at this time and under my present circumstances.

- February 21st 2010, 03:59 PMSoroban
Hello, arrowhead566!

Quote:

How do you inscribe a semicircle into a square?

And the semicircle's diameter can not be the one of the diagonals of the square.

*maximum area*.Code:`C`

- * - - * o * - - - *

: | *:::::|:::::* |

: D o:::::::|:::::::* |

: | *:::::|R:::::::*|

: | R*:::|:::::::::|

x | * |::::R::::*

: | o:-:-:-:-:o B

: | O: *:::::::*

: | : *R::::|

: | : 45°*::*|

- E o - - - * - - - o *

F A

Let = side of the square.

Let = radius of the semicircle.

Let = center of the semicircle.

The semicircle is tangent to the square at

. . Then: .

In right triangle

. . Hence: .

Since , we have: .

. . Hence: .

Therefore: .

- February 21st 2010, 06:42 PMarrowhead566
This is the exact problem.

Inscribe a semicircle into the square given below.

And then my teacher put a picture of what it was supposed to look like, which I have attached in my previous post.

I see what you're doing, but how did you get the semicircle's center? Or what would be the first time in constructing this after the square?

O yes, also when you say what R is equal to, how would you go about constructing a segment that multiplies to side x? - February 21st 2010, 08:23 PMSoroban
Hello again, arrowhead566!

Quote:

When you say: .

how would you go about constructing a segment that multiplies to side ?

On a horizontal line, mark off points

. . so that: .Code:`1 1 1`

- o - - - o - - - o - - - o -

A B C D

At , erect perpendicular so that: .Code:`E`

o _

| * √2

1| *

| *

- o - - - o - - - o - - - o - -

A B C 1 D

Using D as center and DE as radius,

draw an arc cutting BC at P.Code:`E`

. o

. | *

. | *

. | *

- * - - - * o - - * - - - o -

A B P C D

Through , draw a line to the upper-right.

On the line, measure offCode:`*`

*

*

Q *

o

x * \

* \

* \

- o - - - - - o - - o -

A B P

Code:`*`

R *

o

Q * \

o \

x * \ \

* \ \

* \ \

- o - - - - - o - - o -

A B P

Through P, construct

Then: .