Let’s denote the right triangle as \( ABC \) with \( \angle C \) as the right angle. The median to the hypotenuse \( AB \) from vertex \( C \) creates two segments along the hypotenuse \( AD \) and \( DB \) on \( AB \).
Given that the median divides the right angle into two angles in the ratio of \( 1:2 \), we can denote the angles as: \[ \angle ACD = x \quad \text{and} \quad \angle BCD = 2x. \] Since \( \angle ACB \) is a right angle (\( 90^\circ \)), we have the equation: \[ x + 2x = 90^\circ \implies 3x = 90^\circ \implies x = 30^\circ. \] Therefore, we can say the angles are: \[ \angle ACD = 30^\circ \quad \text{and} \quad \angle BCD = 60^\circ. \] This implies that triangle \( ABC \) is a \( 30^\circ-60^\circ-90^\circ \) triangle, which has specific properties regarding the lengths of the sides.
In a \( 30^\circ-60^\circ-90^\circ \) triangle, the ratios of the lengths of the sides opposite these angles are:
- The side opposite \( 30^\circ \) (which we'll denote as \( a \)).
- The side opposite \( 60^\circ \) (which will be \( a\sqrt{3} \)).
- The hypotenuse (opposite \( 90^\circ \)) will be \( 2a \).
Now to determine the perimeter of the triangle \( ABC \): \[ \text{Perimeter} = a + a\sqrt{3} + 2a = a(1 + \sqrt{3} + 2) = a(3 + \sqrt{3}). \]
The given dimensions do not mention the length of the median drawn to the hypotenuse itself, and this is needed for calculating the perimeter explicitly.
However, if we assume the median’s length is given (though not mentioned), the median \( m \) to the hypotenuse for a triangle with side lengths \( a \) and \( b \) while the hypotenuse is \( c \) can be computed as: \[ m = \frac{1}{2} \sqrt{2a^2 + 2b^2 - c^2}. \]
Since \( c^2 = a^2 + b^2 \), we find that: \[ m = \frac{1}{2} \sqrt{2a^2 + 2b^2 - (a^2 + b^2)} = \frac{1}{2} \sqrt{a^2 + b^2} = \frac{1}{2}c. \]
Since \( c = 2a \), we find: \[ m = \frac{1}{2}(2a) = a. \]
If we know the length of the median, we can directly express \( a \) in terms of the median \( m \): \[ m = a. \]
Thus the perimeter becomes: \[ \text{Perimeter} = m(3 + \sqrt{3}) \] where \( m \) is the length of the median.
Without the specific median length provided, it’s impossible to calculate a numerical perimeter. If you had that length, you would substitute it in to find the perimeter as \( m(3 + \sqrt{3}) \).