Question

Points A,B,C,D lie on a line consecutively so that ab=bc=cd. Find the distance between the midway points of the segments ab and cd

Answer 1

Given points \( A, B, C, D \) on a line such that the segments \( AB, BC, \) and \( CD \) are equal in length, we can denote the length of each segment as \( x \).

- Let \( A \) be at position \( 0 \).
- Then point \( B \) will be at position \( x \).
- Point \( C \) will be at position \( 2x \).
- Point \( D \) will be at position \( 3x \).

Next, we need to find the midpoints of segments \( AB \) and \( CD \).

1. **Finding the midpoint of segment \( AB \)**:
\[
\text{Midpoint of } AB = \frac{A + B}{2} = \frac{0 + x}{2} = \frac{x}{2}
\]

2. **Finding the midpoint of segment \( CD \)**:
\[
\text{Midpoint of } CD = \frac{C + D}{2} = \frac{2x + 3x}{2} = \frac{5x}{2}
\]

Now, we need to calculate the distance between the midpoints of segments \( AB \) and \( CD \):
\[
\text{Distance} = \left| \frac{5x}{2} - \frac{x}{2} \right| = \left| \frac{5x}{2} - \frac{1x}{2} \right| = \left| \frac{4x}{2} \right| = \left| 2x \right| = 2x
\]

Thus, the distance between the midpoints of segments \( AB \) and \( CD \) is
\[
\boxed{2x}.
\]