What is the area of an isosceles triangle if two of its sides measure 6 and 12?

Question: What is the area of an isosceles triangle if two of its sides measure 6 and 12?

Solution:
The given triangle is an Isosceles triangle and hence, two of the three sides of the triangle are equal.
Hence, the third side of the triangle can either be 6 or be 12.

If the two equal sides of the triangle measure 6, the sides of the triangle become 6, 6, and 12.
However, the sum of the two smaller sides is not greater than the third side.
∴ 6 is not a possible value of the third side.

If the two equal sides of the triangle measure 12, the sides of the triangle become 6, 12, and 12.
The sum of the smaller two sides is greater than the third side, and hence, the value of the third side is 12.

Let, a = 12, b = 6
∴ Area = (b/4) × √(4a² - b²)
= (6/4) × √{4 × (12)² - (6)²}
= (3/2) × √(576 - 36)
= (3/2) ×√540
= (3/2) × √(36 × 15)
= (3/2) × 6 × √15
= 9√15