What is the area of the largest triangle that can be inscribed in a semi-circle of radius ^@r^@ units?


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Answer:

^@ r^2 \text { sq. units }^@

Step by Step Explanation:
  1. The largest triangle that can be inscribed in a semi-circle with the center ^@ O ^@ will be a right-angled isosceles triangle, ^@ ABC ^@ with ^@ OA = OB = OC ^@ and ^@ OC \perp AB ^@.

    Let us draw the triangle ^@ ABC ^@ inside the semi-circle.
    C A O B

  2. We see that the length of the base of the triangle is equal to the diameter of the circle.

    The radius of the circle = ^@r^@
    So, the base of the triangle = ^@2r^@

    Also, the height of the triangle = ^@r^@

    Thus, the area enclosed by the triangle = ^@ \dfrac { 1 } { 2 } \times Base \times Height = \dfrac { 1 } { 2 } \times 2r \times r = r^2 \text { sq. units. }^@

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