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)

Figure description: A circle with a marked centre. A horizontal chord through the centre (a diameter) forms the base of an inscribed triangle. The two slanted sides of the triangle, going from the two ends of this diameter up to a single apex point on the circle, are labelled 15 and 20. The third (unknown, horizontal) side of the triangle is the diameter of the circle itself. No portion of the figure is shaded/hatched.

• A. 125.50π sq.cm
• B. 134π sq.cm
• C. 156.25π sq.cm
• D. 162.50π sq.cm
• E. 175 sq.cm

Hint:

An angle inscribed in a semicircle is always 90 degrees, so use Pythagoras on the 15-20 legs to get the diameter first.

Answer 1

The Correct Option is C

Step 1: Use the angle-in-a-semicircle property.
Since the unknown side of the triangle is the diameter of the circle, the angle subtended at the apex (the third vertex, opposite this diameter) is inscribed in a semicircle. By the standard circle theorem, any angle inscribed in a semicircle is a right angle. So the triangle is right-angled at the apex, with the two given sides (15 and 20) as the legs, and the diameter as the hypotenuse.

Step 2: Find the diameter using Pythagoras.
\[\text{diameter}^2 = 15^2 + 20^2 = 225 + 400 = 625\]\[\text{diameter} = \sqrt{625} = 25 \text{ cm}\]

Step 3: Find the radius.
\[r = \frac{25}{2} = 12.5 \text{ cm}\]

Step 4: Compute the area of the circle.
\[\text{Area} = \pi r^2 = \pi (12.5)^2 = 156.25\pi \text{ sq.cm}\]

Since the figure has no shaded portion marked anywhere in it, the "unshaded region" being asked about is simply the full area enclosed by the circle, which works out to a clean value because 15, 20, 25 form a Pythagorean triple. This gives the area as 156.25π sq.cm, which is option (c).