Math Problem Statement

Given right triangle ABC with altitude BD drawn to hypotenuse AC: If AD = 12 and DC = 16, what is the length of BD in simplest radical form?

Solution

This problem involves using the geometric mean theorem for a right triangle with an altitude drawn to the hypotenuse. Here’s how to solve it step by step:

Key Information:

  • AD=12AD = 12
  • DC=16DC = 16
  • AC=AD+DC=12+16=28AC = AD + DC = 12 + 16 = 28
  • Altitude BDBD forms two similar right triangles.

According to the geometric mean theorem:

BD=ADDCBD = \sqrt{AD \cdot DC}

Step-by-Step Solution:

  1. Substitute the values of ADAD and DCDC into the formula: BD=1216BD = \sqrt{12 \cdot 16}

  2. Simplify the product: BD=192BD = \sqrt{192}

  3. Simplify 192\sqrt{192} to its simplest radical form: 192=643=83\sqrt{192} = \sqrt{64 \cdot 3} = 8\sqrt{3}

Final Answer:

BD=83BD = 8\sqrt{3}

Would you like further clarification or details about the steps?


Relative Questions:

  1. How does the geometric mean theorem apply to right triangles?
  2. What are other properties of similar triangles in this setup?
  3. How can you simplify square roots systematically?
  4. What are the conditions for triangles to be similar?
  5. How is the hypotenuse related to the legs and altitude in a right triangle?

Tip:

Always verify whether the given lengths satisfy the properties of right triangles (e.g., ACAC as the hypotenuse equals AD+DCAD + DC).

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Math Problem Analysis

Mathematical Concepts

Right triangles
Geometric mean
Simplifying square roots

Formulas

BD = √(AD × DC)

Theorems

Geometric mean theorem

Suitable Grade Level

Grades 8-10