To find the length of \( AD \) in triangle \( ABC \) with the altitude \( CD \), we can use properties of similar triangles and the Pythagorean theorem.
Let \( AB = c \), \( BC = a \), and \( CA = b \).
Since \( CD \) is the altitude, we can denote the segments \( AD = x \) and \( DB = y \).
By the properties of similar triangles, we know that: \[ \frac{AD}{AC} = \frac{CD}{AD} \quad \text{and} \quad \frac{DB}{BC} = \frac{CD}{DB} \]
From these proportions, we can establish the following equations based on the segments:
- \( x^2 = AD \cdot AC \)
- \( y^2 = DB \cdot BC \)
Also, note the relationship between the segments: \[ x + y = c \]
Furthermore, the area of triangle \( ABC \) can be expressed in two ways using the altitude: \[ \text{Area} = \frac{1}{2} \cdot AD \cdot h = \frac{1}{2} \cdot DB \cdot h \]
Where \( h \) is the length of the altitude.
In right triangles, we also have: \[ AD = \frac{AB^2}{AC} \quad \text{and} \quad DB = \frac{BC^2}{AC} \]
Using the Pythagorean Theorem:
Given the right triangle properties, we can apply the Pythagorean theorem: \[ AB^2 = AD^2 + CD^2 \] \[ BC^2 = DB^2 + CD^2 \]
Finally, if you have specific lengths for the sides \( AB \), \( BC \), and \( CA \) of triangle \( ABC \), you can calculate \( AD \). However, I currently do not have those specific lengths. Could you please provide them?
If you have certain variable values or preferences in mind (like \( AD = 2.25 \), \( 2 \), \( 1.75 \), etc.), let me know, so I can assist you further.
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