Question

Given is the right triangle ABC with altitude CD. Apply the Pythagorean Theorem and similar triangles as needed to find the length of AD. Round the answer to two decimal places.
(1 point)

Answer 1

To find the length of segment \( AD \) in the right triangle \( ABC \) with altitude \( CD \), we can use the properties of similar triangles and apply the Pythagorean Theorem. However, I need additional information about the triangle.

1. **Define the lengths**: Let's label:
- \( AC = b \) (the length of one leg),
- \( BC = a \) (the length of the other leg),
- \( AB = c \) (the hypotenuse),
- \( AD = x \),
- \( DB = y \) (hence \( AB = x + y = c \)),
- \( CD = h \) (the altitude from \( C \) to \( AB \)).

2. **Using Similar Triangles**: Triangles \( ACD \), \( BCD \), and \( ABC \) are similar. This gives us the following relationships:
\[
\frac{AD}{AC} = \frac{CD}{BC} \quad \Rightarrow \quad x = \frac{b \cdot h}{a}
\]
\[
\frac{DB}{BC} = \frac{CD}{AC} \quad \Rightarrow \quad y = \frac{a \cdot h}{b}
\]

3. **Rearranging the equation for \( c \)** gives us:
\[
x + y = c \quad \Rightarrow \quad \frac{b \cdot h}{a} + \frac{a \cdot h}{b} = c
\]
If we have \( a \), \( b \), and \( c \), we can express \( h \) in terms of \( a \) and \( b \):
\[
h = \frac{ab}{c}
\]

4. **Plugging in the values**: If you provide the lengths of sides \( a \), \( b \), and \( c \), I can find \( h\) and simplify \( x\) (the length of \( AD\)) accordingly.

If you can provide the specific numerical values for the sides or any additional context about triangle ABC, I would be happy to help you solve for \( AD\) and round the answer to two decimal places!