If three sides of the triangle are given as 32 cm, 34 cm and 34 cm, then the…

Concept

For a triangle with all three sides known, the area equals one-half times the base times the corresponding height. When two sides are equal (isosceles), the perpendicular dropped from the apex to the unequal side bisects that side, so its height can be found by the Pythagoras theorem: (slant side)² = (half-base)² + (height)².

Application

1. The sides are 32 cm, 34 cm and 34 cm. Two sides are equal (34 = 34), so the triangle is isosceles with base 32 cm and equal sides 34 cm.

2. Drop a perpendicular from the apex to the 32 cm base; it bisects the base into two parts of 16 cm each.

3. Apply Pythagoras in one right half: height² = 34² − 16² = 1156 − 256 = 900, so height = 30 cm.

4. Area = ½ × base × height = ½ × 32 × 30 = 16 × 30 = 480 cm².

Cross-check

Heron's formula gives the same result independently. Semi-perimeter s = (32 + 34 + 34) ÷ 2 = 50. Area = √(s(s−a)(s−b)(s−c)) = √(50 × 18 × 16 × 16) = √230400 = 480 cm². Both methods agree, confirming the area is 480 cm².