Final Answer:
The correct value of 'm' based on the theorem's application in this context is 20. Therefore the correct answer is option C.
Step-by-step explanation:
The geometric mean altitude theorem states that in a right triangle, the altitude drawn to the hypotenuse divides the hypotenuse into two segments, and the length of the altitude is the geometric mean of the two segments. Let's denote the segments of the hypotenuse as 'm' and 'n' and the altitude as 'x'. According to the theorem, 'x' is the geometric mean of 'm' and 'n'. Using the formula for the geometric mean (x^2 = m * n), if 'm = 4' and 'n = 5', substituting these values gives us x^2 = 4 * 5 = 20. Therefore, the value of 'x', which represents the altitude, is the square root of 20, which simplifies to x = √20 = 4.472. Among the given options, C. 20 is the accurate value for 'm' in this case.
In this scenario, the theorem's application involves a right triangle with sides 4 and 5 units long. Utilizing the theorem, we find the altitude 'x' by taking the square root of the product of the two segments of the hypotenuse. With 'm = 4' and 'n = 5', substituting into the geometric mean formula (x^2 = m * n) yields x^2 = 4 * 5 = 20. Taking the square root of 20 gives the exact value of the altitude, which is x = √20 = 4.472 units. Hence, the correct value of 'm' based on the theorem's application in this context is 20.
This theorem serves as a fundamental principle in geometry, allowing for the determination of the altitude in a right triangle based on the segments of its hypotenuse. Applying this theorem involves understanding the relationship between the altitude and the segments of the hypotenuse, providing a valuable tool in solving various geometric problems related to right triangles. Therefore the correct answer is option C.