Ask a New Question
Menu

Use the image to answer the 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)

1 answer

To find the length of segment \( AD \) in right triangle \( ABC \) with the given altitude \( CD \), we can use the properties of similar triangles.

Let's denote:

  • Length of \( AC \) as \( b \)
  • Length of \( BC \) as \( a \)
  • Length of \( AB \) (the hypotenuse) as \( c \)

Steps:

  1. Use the Pythagorean Theorem: The Pythagorean Theorem states that \( a^2 + b^2 = c^2 \).

  2. Identify Similar Triangles:

    • Triangles \( ADC \) and \( BDC \) are similar to triangle \( ABC \) because they share angle \( C \) and both have a right angle.
    • Therefore, using the property of similar triangles, we have: \[ \frac{AD}{AC} = \frac{CD}{AB} \] \[ \frac{BD}{BC} = \frac{CD}{AB} \]

  3. Use the formula for the altitude in terms of the legs of the right triangle: The length of the altitude \( CD \) can be calculated using the area relation: \[ CD = \frac{AB \cdot AC \cdot BC}{AB} \]

  4. Finding the length of \( AD \):

    • Given the ratio from similar triangles. Let \( AD = x \). Then \( DB = c - x \).
    • From similarity: \[ \frac{x}{b} = \frac{CD}{c} \] \[ \frac{c - x}{a} = \frac{CD}{c} \]

  5. Solve for \( CD \) using the effective area relation: \[ \text{Area} = \frac{1}{2} \cdot AC \cdot BC = \frac{1}{2} \cdot AB \cdot CD \]

    Rearranging will give us the measurements for calculating \( AD \).

Concluding Calculation:

Now find the lengths based on given measurements, or area derived if numbers are provided.

Once the necessary lengths are plugged in:

  • Calculate \( AD \) as follows, rounding to two decimal places.

Unfortunately, you haven't provided specific lengths or numbers for the triangle sides or the altitude length, needed for numerical calculation. If you could provide those, I'd be able to give you the exact lengths of segments including \( AD \).

Please let me know if you have specific values or measurements!