Question

In the following figure \(\angle\)MNP = 90\(^\circ\), seg NQ \(\perp\) seg MP, MQ = 9, QP = 4, find NQ. 

Hint:

Remember the geometric mean theorem for any right-angled triangle with an altitude to the hypotenuse. The altitude squared is equal to the product of the parts of the hypotenuse. This is a very common problem type in geometry.

Answer 1

Step 1: Understanding the Concept:
In a right-angled triangle, the altitude drawn to the hypotenuse creates two smaller triangles that are similar to the original triangle and to each other. This leads to the geometric mean theorem.

Step 2: Key Formula or Approach:
The theorem of geometric mean states that in a right-angled triangle, the altitude drawn to the hypotenuse is the geometric mean of the two segments it divides the hypotenuse into. \[ NQ^2 = MQ \times QP \]

Step 3: Detailed Explanation:
Given: \[\begin{array}{rl} \bullet & \text{\(\triangle MNP\) is a right-angled triangle with \(\angle MNP = 90^\circ\).} \\ \bullet & \text{NQ is the altitude to the hypotenuse MP.} \\ \bullet & \text{MQ = 9} \\ \bullet & \text{QP = 4} \\ \end{array}\] According to the geometric mean theorem: \[ NQ^2 = MQ \times QP \] Substitute the given values: \[ NQ^2 = 9 \times 4 \] \[ NQ^2 = 36 \] Take the square root of both sides: \[ NQ = \sqrt{36} \] Since length cannot be negative, \[ NQ = 6 \]

Step 4: Final Answer:
The length of NQ is 6.