Calculating Right Triangle Area Using Leg Ratio and Hypotenuse

The problem asks for the area of a right triangle where the lengths of the two legs are in the ratio 5:12, and the hypotenuse is 13 m.

Finding Leg Lengths

Let the lengths of the legs be $5x$ and $12x$, where $x$ is a constant multiplier. According to the Pythagorean theorem, for a right triangle with legs $a$ and $b$ and hypotenuse $c$, we have $a^2 + b^2 = c^2$. Applying this to our triangle:

$ (5x)^2 + (12x)^2 = 13^2 $

Simplify the equation:

$ 25x^2 + 144x^2 = 169 $

Combine the terms:

$ 169x^2 = 169 $

Solve for $x^2$:

$ x^2 = \frac{169}{169} $ $ x^2 = 1 $

Solve for $x$. Since length must be positive, we take the positive square root:

$ x = \sqrt{1} $ $ x = 1 $

Now, find the actual lengths of the legs:

• Leg 1: $5x = 5 \times 1 = 5 \text{ m}$
• Leg 2: $12x = 12 \times 1 = 12 \text{ m}$

Calculating the Area

The area of a right triangle is calculated using the formula: Area = $\frac{1}{2} \times \text{base} \times \text{height}$. The legs of the right triangle serve as the base and height.

$ \text{Area} = \frac{1}{2} \times 5 \text{ m} \times 12 \text{ m} $

$ \text{Area} = \frac{1}{2} \times 60 \text{ m}^2 $

$ \text{Area} = 30 \text{ m}^2 $

Therefore, the area of the steel plate is $30 \text{ m}^2$.