To find the length of segment \( AD \) in the right triangle \( ABC \) with altitude \( CD \) emanating from point \( C \) to hypotenuse \( AB \), we can use the properties of similar triangles and the Pythagorean Theorem.
Step 1: Understand the triangles involved.
Since \( \triangle ABC \) is right-angled at \( C \), we know that the altitude \( CD \) divides the triangle into two smaller right triangles: \( \triangle ACD \) and \( \triangle BCD \).
Step 2: Use the information given.
We are given:
- \( CD = 3 \)
- \( DB = 4 \)
Let \( AD = x \). Since \( D \) partitions \( AB \): \[ AD + DB = AB \] \[ x + 4 = AB \]
Step 3: Apply the geometric mean theorem.
According to the geometric mean theorem for right triangles, we have: \[ CD^2 = AD \times DB \] Substituting the known values: \[ 3^2 = x \cdot 4 \] \[ 9 = 4x \]
Step 4: Solve for \( x \). \[ x = \frac{9}{4} = 2.25 \]
Thus, the length of \( AD \) is: \[ AD = 2.25 \]
Since we need to round our answer to two decimal places, we get: \[ \boxed{2.25} \]
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