Final answer:
The height of the right triangle is 5√2 units.
Step-by-step explanation:
The height of a right triangle can be determined using the Pythagorean theorem, which relates the lengths of the legs of a right triangle to the length of the hypotenuse. In this case, the given angle measures 45 degrees and the base is 10 units. Since the triangle is a right triangle with a 45-degree angle, the other two angles are also 45 degrees, making it an isosceles right triangle.
Since the angle measures are equal, the lengths of the two legs of the triangle will also be equal. Using the Pythagorean theorem, we can set up the equation a² + a² = 10², where 'a' represents the length of the legs.
Simplifying the equation, we get 2a² = 100. Dividing both sides by 2, we get a² = 50. Taking the square root of both sides, we get a = √50, which simplifies to a = 5√2. Therefore, the height of the right triangle is 5√2 units.