The three sides of a triangular plot are in the ratio 5:12:13, and the total distance around its boundary measures 90 cm. Determine the area of this triangle.

Correct Answer: The correct option is 270 cm².

Step-by-Step Explanation:

Step 1: Find the side lengths of the triangle.

The ratio of the three sides of the triangular plot is given as 5 : 12 : 13.

Let the lengths of the sides be:

$a=5x$

$b=12x$

$c=13x$

The total distance around the boundary (perimeter) is 90 cm. Therefore, the sum of all three sides equals 90 cm:

$5x+12x+13x=90$

$30x=90$

$x=\frac{90}{30}$

$x=3$

Substituting $x=3$ back into our expressions for the sides gives:

$a=5\times 3=15\text{ cm}$

$b=12\times 3=36\text{ cm}$

$c=13\times 3=39\text{ cm}$

Step 2: Determine the type of triangle.

We can check if the triangle is right-angled by applying the Pythagorean theorem:

$a^2+b^2=c^2$

Calculating the squares of the side lengths:

${15}^2+{36}^2=225+1296=1521$

${39}^2=1521$

Since the sum of the squares of the two shorter sides equals the square of the longest side, this is a right-angled triangle. The two perpendicular sides are 15 cm (base) and 36 cm (height).

Step 3: Calculate the area of the triangle.

The area of a right-angled triangle is given by the formula:

$\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$

Substituting base = 15 cm and height = 36 cm into the formula:

$\text{Area}=\frac{1}{2}\times 15\times 36$

$\text{Area}=15\times 18$

$\text{Area}=270{\text{ cm}}^2$

Conclusion:

The area of the triangular plot is 270 cm².