Step 1: Understanding the Concept:
We are provided the ratio of the side lengths of a triangle. The numbers 5, 12, and 13 are a very common Pythagorean triple. This instantly tells us the triangle is a right-angled triangle. This property allows us to use the simple area formula for right triangles instead of the more complex Heron's formula.
Step 2: Key Formula or Approach:
1. Pythagorean Theorem test: If $a^2 + b^2 = c^2$, the triangle is right-angled.
2. Area of a right-angled triangle $= \frac{1}{2} \times \text{base} \times \text{height}$.
3. Introduce a scaling factor $x$ for the side ratios.
Step 3: Detailed Explanation:
Let the actual lengths of the sides of the triangle be $5x$, $12x$, and $13x$, where $x$ is a positive constant scale factor.
First, verify it's a right-angled triangle:
Check if $(5x)^2 + (12x)^2 = (13x)^2$:
LHS $= 25x^2 + 144x^2 = 169x^2$
RHS $= (13x)^2 = 169x^2$
Since LHS = RHS, it is a right-angled triangle. The legs forming the right angle are $5x$ and $12x$, and the hypotenuse is $13x$.
The area of this right-angled triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
\[ \text{Area} = \frac{1}{2} \cdot (5x) \cdot (12x) = 30x^2 \]
We are given that the area is exactly 270. Set up the equation:
\[ 30x^2 = 270 \]
Solve for $x^2$:
\[ x^2 = \frac{270}{30} = 9 \]
Taking the principal square root (as side lengths are positive):
\[ x = 3 \]
Now, substitute the value of $x$ back to find the actual lengths of the sides:
Side 1 $= 5x = 5(3) = 15$
Side 2 $= 12x = 12(3) = 36$
Side 3 $= 13x = 13(3) = 39$
The sides of the triangle are 15, 36, and 39.
Step 4: Final Answer:
The sides are 15, 36, 39.