Question

What is the perimeter of the triangle ABC? One of its side is $10\sqrt{3}$ units.
A. BC is the hypotenuse of the right angle triangle ABC.
B. The sum of the areas of the semicircles described on the three sides of the triangle ABC is 100$\pi$ sq. units.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Answer 1

The correct option is C BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Option (C)

Given, sum of areas of semi-circles x, y and z is 100$\pi$

If BC = a cm, AC = b cm, AB = c cm.

Area of semi-circle x is $\frac{\pi }{2}\times \frac{a^2}{4}c{m}^2$

Area of semi circle y is $\frac{\pi }{2}\times \frac{c^2}{4}c{m}^2$

Area of semi circle z is $\frac{\pi }{2}\times \frac{b^2}{4}c{m}^2$

$\frac{1}{2}\frac{\pi }{4}(a^2+b^2+c^2)100\pi$

$a^2+b^2+c^2=800$

From Statement 1: We know that in ABC is a right angle triangle.

$a^2=b^2+c^2=2(b^2+c^2)=800$

$b^2+c^2=400$

$b^2+c^2=a^2$= 400 ;a = 20

b or c = $10\sqrt{3}$ and the sides are 10, $10\sqrt{3}$ , 20

The perimeter = 30 + $10\sqrt{3}$ cm