Find the area of a right-angled triangle whose base is 12 cm and whose hypotenuse is 13 cm. Use the Pythagoras theorem to find the missing perpendicular side before computing the area.

Introduction / Context:
This is a classic right-triangle area question. When you are given the base and hypotenuse, you usually need to find the remaining perpendicular side using the Pythagoras theorem. After that, area is computed using (1/2) * base * height, where 'height' is the perpendicular side to the base.

Given Data / Assumptions:

• Base b = 12 cm
• Hypotenuse c = 13 cm
• Let the perpendicular side be a cm
• Pythagoras: c^2 = a^2 + b^2
• Area = (1/2) * base * perpendicular

Concept / Approach:
Use Pythagoras theorem to find a: a^2 = c^2 - b^2. Then compute area using the right triangle area formula.

Step-by-Step Solution:
c^2 = a^2 + b^2 13^2 = a^2 + 12^2 169 = a^2 + 144 a^2 = 169 - 144 = 25 a = 5 cm Area = (1/2) * 12 * 5 = 30 cm^2

Verification / Alternative check:
The sides 5, 12, 13 form a well-known Pythagorean triple. Since the triangle is consistent, the computed area 30 cm^2 is reliable.

Why Other Options Are Wrong:
40 cm^2: would require height 6.67 cm for base 12, but the computed height is 5 cm. 50 cm^2: would require height 8.33 cm, not possible with hypotenuse 13. 60 cm^2: would require height 10 cm, which would force hypotenuse > 15. 65 cm^2: does not match (1/2)*12*5.

Common Pitfalls:
Using hypotenuse as height directly in area formula (incorrect). Forgetting the 1/2 factor in triangle area. Subtracting in the wrong order when computing a^2 = c^2 - b^2.

Final Answer:
Area = 30 cm^2