Question

if a triangle is inscribed in a circle so that one of the triangles sides is a diameter of the circle, what is the greatest area that the triangle can have in terms of the radius, r, of the circle? the greatest area that the triangle can have in terms of the radius, r, is

Explanation:

Step1: Recall the property of the triangle inscribed in a circle with a diameter as a side

By Thales' theorem, a triangle inscribed in a circle where one side is the diameter is a right - triangle. Let the diameter of the circle be \(d = 2r\) (where \(r\) is the radius of the circle) be the base of the right - triangle. Let the height of the right - triangle be \(h\). The area of a triangle is \(A=\frac{1}{2}\times base\times height\). Here, \(base = 2r\), so \(A=\frac{1}{2}\times(2r)\times h=r\times h\).

Step2: Determine the maximum value of the height

The height \(h\) of the triangle (inscribed in the circle with the diameter as the base) is maximized when the height is equal to the radius \(r\) of the circle. This is because the maximum distance from a point on the circle (other than the endpoints of the diameter) to the diameter is the radius.

Step3: Calculate the area

Substitute \(h = r\) into the area formula \(A=r\times h\). We get \(A = r\times r=r^{2}\).

Answer:

\(r^{2}\)