Question
Use the geometric mean to find x. Round your answer to the nearest tenth. x=_____ (ABCD triangle. AD is 6, DB is 3, CD is x)
Answers
GPT-4o mini
To find \( x \) using the geometric mean in triangle \( ABCD \) where \( AD = 6 \), \( DB = 3 \), and \( CD = x \), we can apply the properties of similar triangles or right triangles, if relevant.
Here, triangles \( ABD \) and \( CDB \) share vertex \( B \), and according to the geometric mean theorem (sometimes known as the altitude theorem), the length of segment \( CD \) can be found using the geometric mean of the two segments \( AD \) and \( DB \).
The formula for the geometric mean in this case is:
\[
x = \sqrt{AD \cdot DB}
\]
Substituting in the values we know:
\[
x = \sqrt{6 \cdot 3}
\]
\[
x = \sqrt{18}
\]
\[
x \approx 4.2426
\]
Rounding to the nearest tenth:
\[
x \approx 4.2
\]
Thus, the value of \( x \) is \( \boxed{4.2} \).
Here, triangles \( ABD \) and \( CDB \) share vertex \( B \), and according to the geometric mean theorem (sometimes known as the altitude theorem), the length of segment \( CD \) can be found using the geometric mean of the two segments \( AD \) and \( DB \).
The formula for the geometric mean in this case is:
\[
x = \sqrt{AD \cdot DB}
\]
Substituting in the values we know:
\[
x = \sqrt{6 \cdot 3}
\]
\[
x = \sqrt{18}
\]
\[
x \approx 4.2426
\]
Rounding to the nearest tenth:
\[
x \approx 4.2
\]
Thus, the value of \( x \) is \( \boxed{4.2} \).