An isosceles triangle has a perimeter of 32 cm. Each of the two equal sides is 5/6 times the base. Find the area of the triangle in square centimetres (sq cm).

Introduction / Context:
This problem combines ratio-based side relations with triangle area. You first determine the base and equal side lengths from the perimeter condition, then use the isosceles triangle property (height splits the base into two equal halves) to compute height using the Pythagoras theorem. Finally, use the standard area formula: (1/2)*base*height. The concept tested is translating fractional side relations into solvable equations and applying right-triangle geometry correctly.

Given Data / Assumptions:

• Perimeter = 32 cm
• Base = b cm
• Each equal side = (5/6)*b cm
• Height from the apex bisects the base into b/2 and b/2

Concept / Approach:
Use perimeter: b + 2*(5/6*b) = 32 to find b. Then compute equal side length. For an isosceles triangle, dropping the altitude creates two right triangles with hypotenuse equal side, one leg b/2, and the other leg = height h. Use: h = sqrt(side^2 - (b/2)^2). Then area = (1/2)*b*h.

Step-by-Step Solution:
Let base = bEqual side = (5/6)*bPerimeter: b + 2*(5/6*b) = 32b + (10/6)*b = 32b + (5/3)*b = (8/3)*b = 32b = 32*(3/8) = 12 cmEqual side = (5/6)*12 = 10 cmHalf base = b/2 = 6 cmHeight h = sqrt(10^2 - 6^2) = sqrt(100 - 36) = sqrt(64) = 8 cmArea = (1/2)*12*8 = 48 sq cm

Verification / Alternative check:
Check perimeter: 10 + 10 + 12 = 32 cm, correct. Since the triangle is isosceles with equal sides 10 and base 12, height 8 is consistent with the 6-8-10 right triangle. So area 48 is reliable.

Why Other Options Are Wrong:
39 and 42 usually come from wrong base calculation or wrong height. 57 and 64 occur if you forget the (1/2) in the area formula or incorrectly compute sqrt(100-36). These do not satisfy the consistent 6-8-10 geometry.

Common Pitfalls:
Misinterpreting “5/6 times the base” as “base is 5/6 of the side.” Forgetting that the altitude bisects the base in an isosceles triangle. Using Pythagoras with the full base (12) instead of half base (6). Dropping the 1/2 factor in triangle area.

Final Answer:
48 sq cm