The perimeter of a rectangle is 84 m. The length of the rectangle is 6 m more than its breadth. A triangle is formed whose base is equal to the diagonal of this rectangle and whose height is equal to the length of the rectangle. Find the area of this triangle in square metres.

Introduction / Context:
This problem connects rectangle properties with triangle area. First, the sides of the rectangle must be determined using the perimeter and the given relation between length and breadth. Then the diagonal of the rectangle becomes the base of a triangle, and the rectangle length becomes the triangle height. Finally, we apply the triangle area formula to find the required value in square metres.

Given Data / Assumptions:

• Perimeter of rectangle = 84 m.
• Length = breadth + 6 m.
• Base of triangle = diagonal of rectangle.
• Height of triangle = length of rectangle.

Concept / Approach:
For a rectangle with length L and breadth B, perimeter P = 2(L + B). With P known and L = B + 6, we can solve for B and L. The diagonal D of the rectangle is found using the Pythagoras theorem: D = sqrt(L^2 + B^2). The triangle area formula is then Area = (1/2) * base * height, where base = D and height = L.

Step-by-Step Solution:
Let breadth = B and length = B + 6.Perimeter: 2(L + B) = 84.So L + B = 42.Substitute L = B + 6: B + 6 + B = 42, so 2B + 6 = 42.2B = 36, hence B = 18 m.Then L = B + 6 = 24 m.Diagonal D = sqrt(L^2 + B^2) = sqrt(24^2 + 18^2) = sqrt(576 + 324) = sqrt(900) = 30 m.Triangle base = 30 m and height = 24 m.Area of triangle = (1/2) * 30 * 24 = 15 * 24 = 360 sq metre.

Verification / Alternative check:
A quick check confirms the numbers: a 18 m by 24 m rectangle has perimeter 2*(18 + 24) = 84 m, which matches the question. The diagonal of a 18-24 rectangle is a classic 3-4-5 scaled triangle with sides 18, 24, 30. Thus the diagonal of 30 m is correct. Using base 30 and height 24 gives area 360 sq metre, which is consistent.

Why Other Options Are Wrong:
400 sq metre and 420 sq metre result from misusing base or height or forgetting the half factor. 380 sq metre suggests an arithmetic slip. The option 360 metre is dimensionally wrong because it lacks square units, so it cannot represent area.

Common Pitfalls:
Students often forget that perimeter is 2(L + B) and incorrectly use L + B = 84. Another common error is to use L or B as the diagonal instead of applying the Pythagoras theorem. Some learners also forget the factor 1/2 in the triangle area formula, which doubles the area incorrectly. Always check units: area must be in square metres, not just metres.

Final Answer:
360 sq metre