Answered

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Given any two segments of length a and b , it is always possible to construct a segment whose length is the arithmetic mean (average) of a and b . Is it true or false

Sagot :

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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.

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