What is the radius of the circle, in centimeters? [Please round up to the nearest centimeter]?

Question

In the following diagram, each blue line is 24 centimetres long, and the red line is 16 centimetres long. Assume the shape in the centre of the circle is a perfect square, and it is exactly centred in the circle. Also assume that all three coloured lines are parallel to each other.
picture below:
http://i70.photobucket.com/albums/i103/azian_dreamz/217_squarecircle.gif

Answer 1

Norrie presumes that the other ends of the blue lines and the other end of the red line are collinear. This isn't quite right.

First, draw a line connecting the diagonal of the square that touch the ends of the blue lines. Now, extend the red line until it intersects this diagonal. The center of the circle and the center of the square are the same point.
Because the blue lines are parrallel and the same length and because the square is given to be centered in the circle, that means that the blue lines are both perpendicular to the diameter of the circle that goes through the corners touching the blue lines. Again, since the red line is given as parrallel to the blue lines, the extension of the red line will be perp. to the same diameter.

So, the distance from the center of the circle out to where the red line hits the corner of the square is the same as the distance over to where the blue line hits the corner of the square. This distance, when added to the length of the red line, gives the radius of the circle and can be used to find the circumference by C = 2 pi r.

Now, draw a line from the center of the circle over to where either blue line hits the edge of the circle. This forms a right triangle whose hypotenuse is the radius, the base is the distance 'x' from the center to the corner where the blue line hits the square, and the other leg is given to be of length 24.

So, 24^2 + x^2 = (x + 16)^2

or, 576 + x^2 = x^2 + 32x + 256

32x = 320, or x = 10

This means that the radius is 16 + 10 = 26

The circumference must be 52 pi units.

Answer 2

I will do it symbolically. Let the red line be a, the blue line be b, (knowns); let the radius be r and the square's diagonal be d (unknowns). We can set the following relations:

Extend the red line to the center of the square. This becomes a radius, and is the sum of half the square's diagonal plus a, so

d/2+a=r. or d/2 = r - a

Drop a perpendicular from where blue intersects the circle to the red line. Draw a diagonal of the square from its center to the corner that starts the blue line. This diagonal is perpendicular to the red line, and its length is equal to the perpendicular dropped from the circle. The length of this latter perpendicular is then d/2: it forms a right triangle with the circle's radius (drawn to the point of intersection of blue line and circle) and the red line. The length of the leg of this triangle that lies along the red line is b. Using the Pythagorean theorem.

r^2 = (d/2)^2 + b^2.

Substitute for d/2 from the first equation:

r^2 = (r - a)^2 + b^2

Expand the binomial containing r:

r^2 = r^2 = -2*a*r + a^2 + b^2

0 =-2*a*r + a^2 + b^2

r = (b^2 + a^2)/ 2*a

For your numbers, r = 26

Answer 3

Draw a diagonal across the square from blue to blue.
Then, extend the red line to the diagonal line. This will be the centre of the circle..
Making the red line the same length as the blue lines.
Therefore the radius will be 24cm.