Use the image to answer the question.

Question

On right triangle upper A upper C upper B, the altitude of vertex upper C is drawn as a solid line that extends from upper C to the hypotenuse, upper A upper B. The intersection of the altitude and upper A upper B is labeled upper D.

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 \( AD \) in the right triangle \( ABC \) with altitude \( CD \), we can use the properties of similar triangles and the Pythagorean theorem.

1. Identify Similar Triangles:
   In triangle \( ABC \), the altitude \( CD \) creates two smaller triangles: triangle \( ACD \) and triangle \( BCD \). Both of these smaller triangles are similar to triangle \( ABC \) because they all share an angle (the right angle) and have one angle that is common to the corresponding triangles.

2. Use the Pythagorean Theorem:
   Let:

   • \( AB = c \) (the length of the hypotenuse),
   • \( AC = b \),
   • \( BC = a \),
   • \( AD = x \),
   • \( DB = y \).

   Since \( D \) lies on \( AB \), we have: \[ AD + DB = AB \implies x + y = c. \]

3. Apply the Relationship between Parts and the Whole: Since triangles \( ACD \), \( BCD \), and \( ABC \) are similar, we can write the following ratios: \[ \frac{AD}{AB} = \frac{AC}{BC} \implies \frac{x}{c} = \frac{b}{a}, \] which leads to: \[ x = \frac{b}{a} c. \]

4. Substitute Values:
   If specific lengths of sides \( a \), \( b \), and \( c \) are given, we can compute \( x \) directly. Since we don't have those specifics here, we would need numerical values.

5. Finalize Calculation and Round:
   Once you find \( x \), ensure that you round your answer to two decimal places for the final result.

Without specific lengths provided, I cannot give a numerical answer. If you can provide the lengths of sides \( a \), \( b \), and \( c \), I would calculate \( AD \) for you.