The correct answer is option (C):
84 sq.cm
Let's analyze this geometry problem. We are told a triangle has a height and three sides that are consecutive integers. We can represent these dimensions as x, x+1, x+2, and x+3, where x is the smallest of these integers. Also, the dimensions must be measurable using only a 15 cm measurement scale, which likely means we are measuring in whole cm units.
We know the area of a triangle can be calculated as (1/2) * base * height. In our case, the height will be one of our integers, and a side will be the base. Let's consider the possible scenarios and test the given answer choices. We will need to test the area calculations, keeping in mind that we want whole number dimensions, using a measurement scale of 15cm.
Since the area is given in square centimeters, we can start by considering the answer choices.
Option 1: 64 sq.cm
If the height is x and the base is x+1, then (1/2) * x * (x+1) = 64. So x(x+1) = 128. We can see that 11*12 = 132 and 10*11 = 110, so x would not be an integer. We also know that x would need to be less than 15.
Option 2: 72 sq.cm
If the height is x and the base is x+1, then (1/2) * x * (x+1) = 72. So x(x+1) = 144. Then x = 11, x+1 = 12, so the sides are 11,12,13,14, and Area = (1/2)*12*11 = 66, Not 72.
Lets try height as x, base as x+2; (1/2) * x * (x+2) = 72. x(x+2) = 144. x^2 + 2x - 144 = 0, no integer solutions
Option 3: 84 sq.cm
We can test a few cases.
Case 1: Height = 12, Base = 14. Area = (1/2)*12*14=84. Sides are 11, 12, 13, 14. If the height is 12 and the base is 14. The sides include 12 and 14, and the other two sides would have to be either 11 and 13 (or something different but the dimensions are not necessarily equal). This is a possible solution using integers and the given consecutive condition.
Case 2: If we assume that height is x, base is x+1, (1/2)*x*(x+1) = 84. x(x+1) = 168. x^2+x-168=0, no integer solutions.
Case 3: if height is x, base x+2, (1/2) * x * (x+2) = 84. x(x+2)=168. x^2 +2x -168 =0, no integer solution.
Case 4: If height is x, base x+3, (1/2)*x*(x+3) = 84. x(x+3) = 168. x^2 +3x -168 =0, no integer solutions.
Option 4: 96 sq.cm
(1/2) * x * (x+1) = 96. x(x+1) = 192. 13*14 = 182 and 14*15 = 210, so no integer solution.
Try base to be x+2. (1/2)*x*(x+2) = 96. x(x+2) = 192. x^2+2x -192=0, no integer solutions.
Option 5: 108 sq.cm
(1/2) * x * (x+1) = 108. x(x+1) = 216. 14*15 is close, not an integer solution.
Try base as x+2. (1/2) * x * (x+2) = 108. x(x+2) = 216. No integer solution.
Only Option 3 gave us an integer solution for the dimensions of 11,12,13,14. Also the maximum dimension is 14, within the 15cm measurement scale.
Therefore, the area of the triangle is 84 sq.cm.