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It a right triangle is inscribed in a circle the the what is a diameter of the circle?
The diameter is the distance across the centre of the circle.
When a triangle with one side length the diameter of a circle is inscribed in that circle does it have to be a right triangle?
Yes. It follows from one of the circle theorems which states that the angle subtended in a semicircle is a right angle.
What theorem did Thales gave to geometry?
The theorem where a triangle inscribed in a circle is right if and only if one of the legs is a diameter.
How would you construct a right triangle given the length of a leg and the radius of the circumscribed circle?
To construct a right triangle given the radius of the circumscribed circle and the length of a leg, begin with two ideas. First, the diameter of the circle is equal to twice the radius. That's pretty easy. Second, the diameter of the circle is the length of the hypotenuse. The latter is a key to construction. Draw your circle, and draw in a diameter, which is the hypotenuse of the right triangle, as was stated. Now set you compass for the length of the leg of the triangle. With this set, place the point of the compass on one end of the diameter (the hypotenuse of your triangle), and draw an arc through the circumference of the circle. The point on the curve of the circle where the arc intersects it will be a vertex of your right triangle. All that remains is to add the two legs or sides of the triangle. Draw in line segments from each end of the hypotenuse (that diameter) to the point where your arc intersected the curve of the circle. You've constructed your right triangle. Note that any pair of lines that is drawn from the ends of the diameter of a circle to a point on the curve of the circle will create a right triangle.
How do you work out the radius of a circle from a chord?
This is hard to explain without a diagram, but draw a circle with a chord through it. Now draw a line from the center of the circle to the center of the chord and you'll have a right angle. Draw the hypotenuse of this new right triangle, which just happens to be the radius. I took geometry last year, so I'm kind of stuck at this point, but I know you have to work with that right triangle.
