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.