Question

In the figure below, the unknown side of the triangle is the diameter of the circle. What is the area of the unshaded region? (Figure not drawn to scale)

• A. \(125.50\pi\) sq.cm
• B. \(134\pi\) sq.cm
• C. \(156.25\pi\) sq.cm
• D. \(162.50\pi\) sq.cm

Hint:

A triangle inscribed in a circle with one side equal to the diameter is always right-angled at the opposite vertex (angle in a semicircle).

Answer 1

The Correct Option is C

Step 1: Identify the right angle.
The unknown side of the triangle is the diameter of the circle. By the angle-in-a-semicircle theorem, the angle opposite the diameter (at the vertex where the sides 15 and 20 meet the circle) is \(90^\circ\), so the triangle is right-angled with legs 15 cm and 20 cm and the diameter as its hypotenuse.
Step 2: Find the diameter.
\(diameter = \sqrt{15^2+20^2} = \sqrt{225+400} = \sqrt{625} = 25\) cm.
So the radius \(r = 12.5\) cm.
Step 3: Identify the unshaded region.
The figure only shows the triangle drawn inside the circle with no portion marked or hatched, so the entire circular region is the unshaded region; there is nothing subtracted from it.
Step 4: Compute the circle's area.
\(Area = \pi r^2 = \pi (12.5)^2 = 156.25\pi\) sq.cm.\[\boxed{Unshaded\ area = 156.25\pi\ sq.cm}\]