Points $$ A, B, C $$, and $$ D $$ are on a line such that $$ B $$ is between $$ A $$ and $$ C $$, and $$ C $$ is between $$ B $$ and $$ D $$. The distance from $$ A $$ to $$ B $$ is 6 units. The distance from $$ B $$ to $$ C $$ is twice the distance from $$ A $$ to $$ B $$, and the distance from $$ C $$ to $$ D $$ is twice the distance from $$ B $$ to $$ C $$. What is the distance, in units, from the midpoint of $$ \overline{B C} $$ to the midpoint of $$ \overline{C D} $$ ?

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Let's assign coordinates to the points. Suppose A is at 0. Then, since AB = 6, point B is at 6. The problem states that BC is twice AB, so BC = 12; hence, point C is at 6 + 12 = 18. Next, CD is twice BC, so CD = 24; thus, point D is at 18 + 24 = 42.

Now, find the midpoints:
• The midpoint of BC is (6 + 18) ÷ 2 = 12.
• The midpoint of CD is (18 + 42) ÷ 2 = 30.

The distance between these two midpoints is 30 − 12 = 18 units.

Therefore, the distance from the midpoint of segment BC to the midpoint of segment CD is 18 units.
The distance from the midpoint of segment BC to the midpoint of segment CD is 18 units.

Points , and are on a line such that is
between and , and is between and . The
distance from to is 6 units. The distance from
to is twice the distance from to , and the
distance from to is twice the distance from to
. What is the distance, in units, from the midpoint
of to the midpoint of ?

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Points , and are on a line such that is between and , and is between and . The distance from to is 6 units. The distance from to is twice the distance from to , and the distance from to is twice the distance from to . What is the distance, in units, from the midpoint of to the midpoint of ?