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.

Answer 1

To solve the problem, we need to use the concepts related to right triangles, specifically the Pythagorean theorem and properties of similar triangles.

Let's denote:
- \( AB = c \)
- \( BC = a \)
- \( AC = b \)
- \( CD = h \) (altitude from C to AB)
- \( AD = x \)
- \( DB = y \)

In triangle ABC, since it is a right triangle, the Pythagorean theorem states:

\[
AB^2 = AC^2 + BC^2 \quad \text{or} \quad c^2 = a^2 + b^2
\]

We'll also use the fact that the altitude from the right angle to the hypotenuse creates two smaller triangles, ACD and BCD, which are similar to triangle ABC. The relationships that come from the similarity are:

1. \(\frac{AD}{AC} = \frac{CD}{AB}\)
2. \(\frac{DB}{BC} = \frac{CD}{AB}\)

From the similarity, we can express \(AD\) and \(DB\) in terms of the segments of the hypotenuse.

The length of the altitude \(CD\) can be calculated using the formula:

\[
CD = \frac{AB \cdot AC}{BC} = \frac{c \cdot b}{a}
\]

Using the similarity ratios:

\[
AD = \frac{AC^2}{AB}
\]
\[
DB = \frac{BC^2}{AB}
\]

Since \(AD + DB = AB\), we can write:

\[
x + y = c
\]

Now, substituting these values:

\[
\frac{b^2}{c} + \frac{a^2}{c} = c
\]

This leads us to:

\[
x = \frac{b^2}{c}
\]

Thus, if you need both lengths, you can calculate \(x\) and \(y\) as needed.

To find length \(AD\):
1. You substitute the known lengths into the formula.
2. Round your result to two decimal places.

If you've provided specific values for \(a\), \(b\), \(c\), we can compute \(AD\) directly. Please provide the lengths of the triangle sides or any additional information if you'd like me to calculate a specific example.