Answer:
The statement is true.
Step-by-step explanation:
1. Definition of Arithmetic Mean:
The arithmetic mean (average) of two numbers ( a ) and ( b ) is given by:Mean=2a+b
2. Constructing the Segment:
Given two segments of lengths ( a ) and ( b ), we can always find their arithmetic mean using the formula above.
Example:
Suppose ( a = 4 ) and ( b = 6 ).
The arithmetic mean is: Mean=24+6=210=5
3. Constructing the Segment of Length 5:
It is always possible to construct a segment of any given length using standard geometric tools (like a ruler or compass). Therefore, a segment of length 5 can be constructed.
General Case:
For any lengths ( a ) and ( b ), the arithmetic mean ( \frac{a + b}{2} ) is a positive real number (assuming ( a ) and ( b ) are positive).
Since we can always construct a segment of any positive real length, it is always possible to construct a segment whose length is the arithmetic mean of ( a ) and ( b ).
Thus, the statement is true.