Multiple choice

A triangle is inscribed in a semi circle of radius $a$ so that one side is the bounding diameter. Then the maximum area of the triangle is

  1. $(\pi/4){a}^{2}$
  2. $(1/2){a}^{2}$
  3. $(1/4){a}^{2}$
  4. ${a}^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a triangle inscribed in a semicircle with the diameter as one side, the height is maximized when the vertex is at the radius perpendicular to the diameter. Max height = radius = a. Base = diameter = 2a. Area = (1/2) * base * height = (1/2) * 2a * a = a^2.

AI explanation

Let the bounding diameter lie on the x-axis with endpoints (-a,0) and (a,0), and let the third vertex be (a*cos(t), a*sin(t)). The base of the triangle is the diameter, which has a length of 2a, and the height is the y-coordinate of the third vertex, a*sin(t). The area is (1/2) * base * height = (1/2) * 2a * a*sin(t) = a^2*sin(t). The maximum value of sin(t) is 1, so the maximum area of the triangle is a^2.

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