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 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 to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse, and .
The formula is:
From the given values:
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.Ask a new question for Free
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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