# A 34 cm segment AB was divided into three unequal segments AC, CD, DB.

A 34 cm segment AB was divided into three unequal segments AC, CD, DB. The distance between the midpoints of the extreme segments is 20cm. Find the length of the segment CD.

Pst x cm is the length of half of the segment AC, y cm is the length of half of the segment DB, z cm is the length of the segment CD. (x + y + x) cm is the length from the middle of the segment AC to the middle of the segment DB, according to the condition, the length between the midpoints of these segments is 20 cm. Let’s compose the first equation: x + y + z = 20. The length of the entire segment is written as follows: ( 2x + 2y + z) cm. By condition, the length of the entire segment is 34 cm. Let’s compose the second equation: 2x + 2y + z = 34. From two equations we will compose the system:

x + y + z = 20;

2x + 2y + z = 34;

x + y + z = 20;

2 * (x + y) + z = 34;

We introduce a change of variables: x + y is denoted by the variable p:

p + z = 20;

2p + z = 34;

z = 20 – p;

z = 34-2p;

Let’s equate the right-hand sides of the equations:

20 – p = 34 – 2p;

p = 34-20;

p = 14;

z = 20-14;

z = 6 (cm) – the length of the CD segment.