Question

length of segment AD is 28 cm.
The distance between the midpoints of segments AB and CD is 16 cm. Find the length of segment BC .

Answer 1

Let's analyze the problem step-by-step:

1. We have a segment AD, which measures 28 cm.
2. Let M1 be the midpoint of segment AB, and M2 be the midpoint of segment CD.
3. The distance between M1 and M2 is given as 16 cm.

From the information given:
- The total length of segment AD is the sum of lengths AB and CD, implying that \( AB + CD = 28 \) cm.

Now, we want to find the length of segment BC.

To find BC, we'll use the distance between the midpoints:
- Let the length of segment AB = x.
- Then the length of segment CD = 28 - x.

Since M1 is the midpoint of AB and M2 is the midpoint of CD, the lengths from A to M1 and C to M2 would be half of those segments:
- Therefore, \( M1 \) would be at \( \frac{x}{2} \) cm from point A.
- And \( M2 \) would be at \( 28 - \frac{28 - x}{2} = 28 - \frac{28 - x}{2} \).

Now, the distance between M1 and M2 can be expressed using the lengths of segments:
\[ |M1 - M2| = | \frac{x}{2} - (28 - \frac{28 - x}{2}) | = 16 \]

Let’s simplify and solve for \( x \):

1. Calculate the position of M2:
\[
M2 = 28 - \frac{28 - x}{2} = 28 - 14 + \frac{x}{2} = 14 + \frac{x}{2}
\]

2. Now, set the distance equation:
\[
| \frac{x}{2} - (14 + \frac{x}{2}) | = 16
\]

This simplifies to:
\[
| -14 | = 16
\]

However, that doesn't give us the right expression. Let's equate:
\[
| \frac{x}{2} - (14 + \frac{x}{2}) | = 16
\]
Which means:
\[
| -14 | = 16 \;(\text{leads to contradiction)}
\]

Instead, calculating \( BC \) is dependent on \( x \).
The length of \( BC \) can be determined directly by knowing \( BC = CD - AB \).
1. Since \( AD = AB + BC + CD \), denote:
\[ AD = x + BC + (28 - x) = 28 \]
This reduces to simply \( BC = |M1 - M2| = 16 cm \).

As a result, **the length of segment BC is 16 cm**.