Question
Use the geometric mean to find x. Round your answer to the nearest tenth. x=_____ (ABCD triangle. AD is 6, DB is 3)
Answers
GPT-4o mini
To find the value of \( x \) using the geometric mean in the context of triangle \( ABCD \) where \( AD \) is 6 and \( DB \) is 3, we can use the property of segments in similar triangles.
In triangle \( ADB \), if we let \( AB \) be the hypotenuse and \( AD \) and \( DB \) be the segments from point \( D \) to \( A \) and \( B \), respectively, the geometric mean can be calculated using the formula:
\[
x = \sqrt{AD \cdot DB}
\]
Substituting the values for \( AD \) and \( DB \):
\[
x = \sqrt{6 \cdot 3} = \sqrt{18}
\]
Calculating \( \sqrt{18} \):
\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]
Using the approximate value of \( \sqrt{2} \approx 1.414 \):
\[
3\sqrt{2} \approx 3 \cdot 1.414 \approx 4.242
\]
Rounding to the nearest tenth, we get:
\[
x \approx 4.2
\]
So, the final answer is:
\[
x \approx 4.2
\]
In triangle \( ADB \), if we let \( AB \) be the hypotenuse and \( AD \) and \( DB \) be the segments from point \( D \) to \( A \) and \( B \), respectively, the geometric mean can be calculated using the formula:
\[
x = \sqrt{AD \cdot DB}
\]
Substituting the values for \( AD \) and \( DB \):
\[
x = \sqrt{6 \cdot 3} = \sqrt{18}
\]
Calculating \( \sqrt{18} \):
\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]
Using the approximate value of \( \sqrt{2} \approx 1.414 \):
\[
3\sqrt{2} \approx 3 \cdot 1.414 \approx 4.242
\]
Rounding to the nearest tenth, we get:
\[
x \approx 4.2
\]
So, the final answer is:
\[
x \approx 4.2
\]