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.
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.
Hello, arrowhead566!
I believe that the original problem asked for the semicircle of maximum area.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.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: .
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?
Hello again, arrowhead566!
I've come up with a construction; maybe someone can improve on it.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: .DrawCode: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.Hence: .Code:E . o . | * . | * . | * - * - - - * o - - * - - - o - A B P C D
Through , draw a line to the upper-right.
On the line, measure offDrawCode:* * * 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: .