Final answer:
The base of the triangle is double its height, and the area is 12 cm². The perimeter is the sum of the three sides, and the diagonal is calculated using the Pythagorean theorem.
Step-by-step explanation:
Let's assume that the height of the right triangle is 'h' units. Since the base measures double the height, the base would be '2h' units.
Now, let's use the formula for the area of a triangle: A = 1/2 * base * height. Substituting the values, we have 12 = 1/2 * 2h * h.
Simplifying the equation, we get 12 = h^2. Taking the square root of both sides, we find that h = sqrt(12) = 2sqrt(3).
Now that we know the height, we can find the base, which is 2h = 2 * 2sqrt(3) = 4sqrt(3).
To calculate the perimeter, we need to sum up the lengths of all three sides. The two legs (base and height) are 4sqrt(3) and 2sqrt(3), respectively, while the hypotenuse (diagonal) can be found using the Pythagorean theorem: c^2 = a^2 + b^2. Letting 'c' represent the hypotenuse, we have c^2 = (4sqrt(3))^2 + (2sqrt(3))^2 = 48 + 12 = 60. Taking the square root of both sides, c = sqrt(60) = 2sqrt(15).
Therefore, the perimeter of the triangle is 4sqrt(3) + 2sqrt(3) + 2sqrt(15), and its diagonal (hypotenuse) is 2sqrt(15).
Learn more about Calculating perimeter and diagonal of a right triangle