ΔABC is a right-angled triangle with ∠ABC = 90°. If m(AB) = 28 cm, and m(BC) = 96 cm, find the area (in cm²) of the circumcircle of ΔABC. (Use π = 3.14.)
- (a)7,536
- (b)7,693
- (c)8,164
- (d)7,850
Answer
Why
Correct — D. The right angle is at B, so the side opposite it, AC, is the hypotenuse.
AC² = 28² + 96² = 784 + 9216 = 10000
AC = 100 cm
In a right-angled triangle the hypotenuse is a diameter of the circumcircle, so r = 100 ⁄ 2 = 50 cm.
Area = πr² = 3.14 × 50² = 3.14 × 2500 = 7850 cm² → option (d).
Why the others are wrong
- (a)7,536 — 7,536 is 3.14 × 2400, so it assumes r² = 2400 and a radius near 48.99. The hypotenuse comes out exactly 100, making the radius exactly 50.
- (b)7,693 — 7,693 is 3.14 × 2450, which would need a hypotenuse of about 98.99 cm. Here 28² + 96² is the perfect square 10000, so nothing is rounded.
- (c)8,164 — 8,164 is 3.14 × 2600, the area for a radius of about 50.99 cm. Halving the hypotenuse gives 50 exactly, not a shade over it.
Concept
Two standard results meet here. The right angle at B makes AC the hypotenuse, and 28-96-100 is the 7-24-25 Pythagorean triple scaled by 4, so no long square-rooting is needed.
The result being tested is the second one: in a right-angled triangle the hypotenuse is a diameter of the circumcircle. The circumcentre is therefore the midpoint of the hypotenuse and R = hypotenuse ⁄ 2.
That converts a triangle question into a circle question in one line, and what remains is πr² using the π the paper supplies.
The paper supplies π = 3.14, not 22⁄7. Use what it gives: 22⁄7 × 2500 is about 7857, which is not an option but sits close enough to 7,850 to make a candidate doubt a correct answer.
Key facts
- In a right-angled triangle the hypotenuse is a diameter of the circumcircle, so the circumradius is half the hypotenuse.
- 28, 96, 100 is the 7-24-25 Pythagorean triple multiplied by 4.
- In the notation angle ABC, the vertex is the middle letter B, so the side opposite it is AC.
- With π = 3.14 and r = 50, the circle's area is 3.14 × 2500 = 7850 cm².
Study next
Common traps
- Treating BC = 96 as the hypotenuse because it is the longer given side.
- Finding the circle's circumference, 2πr = 314 cm, instead of its area.
- Answering with the triangle's own area, ½ × 28 × 96 = 1344 cm².
Spotting the triple is the first step, not the answer — SSC hangs a second figure off the right angle.
On 17 Sep 2024, 12:30, Quant Q.11 a perpendicular CM is dropped onto the hypotenuse of a right-angled triangle; on 23 Sep 2024, 09:00, Quant Q.6 you are asked only which of four sets of sides is right-angled at all.
Related PYQs
No directly related past PYQ was found.