Introduction / Context:
When three sides are known, Heron’s formula is the standard method to compute area without explicitly finding heights or angles.
Given Data / Assumptions:
- Sides: a = 26 cm, b = 24 cm, c = 10 cm.
- All satisfy triangle inequality (sum of any two exceeds the third).
Concept / Approach:
Heron’s formula: Area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c)/2 is the semiperimeter.
Step-by-Step Solution:
s = (26 + 24 + 10)/2 = 60/2 = 30.Area^2 = 30 * (30 − 26) * (30 − 24) * (30 − 10) = 30 * 4 * 6 * 20.Compute: 30 * 4 = 120; 6 * 20 = 120; 120 * 120 = 14,400.Area = √14,400 = 120 sq. cm.
Verification / Alternative check:
A 10–24–26 is not a right triangle (since 10^2 + 24^2 = 100 + 576 = 676 = 26^2), actually it is right-angled! Thus area = (1/2)*10*24 = 120, confirming Heron’s result.
Why Other Options Are Wrong:
108, 112, 116 do not match √14,400 and also contradict the right-triangle shortcut result.
Common Pitfalls:
Arithmetic slips in multiplying the four terms or taking the square root.
Final Answer:
120 sq.cm
Discussion & Comments