Math Problem Statement

Given right triangle ABC with altitude BD drawn to hypotenuse AC. If AD = 16 and DC = 44, what is the length of BD in simplest radical form?

Solution

To find the length of segment BDBD in the given right triangle, we can use the geometric mean theorem (also known as the altitude-on-hypotenuse theorem). This theorem states that the length of the altitude BDBD to the hypotenuse ACAC is the geometric mean of the lengths of the two segments of the hypotenuse, ADAD and DCDC.

The formula is:

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

From the given values:

AD=16andDC=44AD = 16 \quad \text{and} \quad DC = 44

Now, we can plug these values into the formula:

Next, let's simplify $$\sqrt{704}$$: \[ \sqrt{704} = \sqrt{16 \times 44} = \sqrt{16} \times \sqrt{44} = 4\sqrt{44}$$ Now, we simplify $$\sqrt{44}$$: $$\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}$$ So, the length of $$BD$$ is: $$BD = 4 \times 2\sqrt{11} = 8\sqrt{11}$$ Therefore, the length of $$BD$$ is $$8\sqrt{11}$$ in simplest radical form. Let me know if you'd like further clarification or additional details! Here are five related questions for you: 1. How does the altitude-on-hypotenuse theorem apply in other geometric scenarios? 2. Can the geometric mean theorem be used in triangles that are not right triangles? 3. What would happen if we used the Pythagorean theorem here instead of the geometric mean theorem? 4. How can we prove that the geometric mean theorem works using similar triangles? 5. What are other methods for finding the length of an altitude in a triangle? **Tip:** Always look for patterns and known theorems like the geometric mean theorem to simplify geometric problems.

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Math Problem Analysis

Mathematical Concepts

Geometry
Right Triangle
Altitudes
Geometric Mean

Formulas

BD = √(AD × DC)

Theorems

Geometric Mean Theorem (Altitude-on-Hypotenuse Theorem)

Suitable Grade Level

Grades 9-12