To solve this problem, we can follow these steps:

Step 1: Assign variables

Let the shorter leg be $x$. Then the longer leg is $2x$, as one leg is twice as long as the other. The hypotenuse is given as $15 \, \text{cm}$.

Step 2: Use the Pythagorean Theorem

The Pythagorean Theorem states: $x^2 + (2x)^2 = 15^2$ Simplify the equation: $x^2 + 4x^2 = 225$ $5x^2 = 225$

Step 3: Solve for $x^2$

$x^2 = \frac{225}{5} = 45$

Step 4: Find $x$

$x = \sqrt{45} = 3\sqrt{5} \, \text{cm} \, \text{(shorter leg)}$ $2x = 2 \times 3\sqrt{5} = 6\sqrt{5} \, \text{cm} \, \text{(longer leg)}.$

Step 5: Calculate the area of the triangle

The area of a right triangle is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}.$ Here, the base and height are the two legs: $\text{Area} = \frac{1}{2} \times x \times 2x = \frac{1}{2} \times (3\sqrt{5}) \times (6\sqrt{5}).$

Simplify: $\text{Area} = \frac{1}{2} \times 18 \times 5 = \frac{90}{2} = 45 \, \text{cm}^2.$

Final Answer:

The area of the triangle is 45 cm².

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Would you like more details or have any other questions?

Related Questions:

1. How can the hypotenuse of a triangle be calculated if both legs are known?
2. How does the Pythagorean theorem apply to non-right triangles?
3. What is the relationship between the area and the perimeter of a right triangle?
4. Can the same problem be solved geometrically instead of algebraically?
5. How would the area change if the hypotenuse were scaled proportionally?

Tip:

In right triangles, the Pythagorean theorem is a powerful tool to find missing sides. Always identify the legs and hypotenuse before applying it.