# In the trapezoid, a segment is drawn parallel to the bases and dividing it into two trapezoids

**In the trapezoid, a segment is drawn parallel to the bases and dividing it into two trapezoids of the same area. Find the length of this segment if the bases of the trapezoid are equal to 24√2 and 7 √2 cm.**

Determine the area of the trapezium KВСM.

Skvsm = (ВС + КМ) * СО / 2.

Let us determine the area of the trapezoid AKMD.

Sacmd = (KM + AD) * HO / 2.

By condition, these areas are equal.

(BC + KM) * CO / 2 = (KM + AD) * HO / 2.

(BC + KM) / (KM + AD) = HO / CO. (1).

Let’s draw a straight line through point M, parallel to AB. Point E is the intersection of this straight line with BC, and point K with AD.

Triangles CKM and DKM are similar in two angles.

Length CE = (KM – BC), length DK = (AD – KM).

Then DK / CE = HO / CO, since HO and CO are the heights of these triangles.

Then HO / CO = (AD – KM) / (KM – BC).

Substitute in equation 1.

(BC + KM) / (KM + AD) = (AD – KM) / (KM – BC).

(7 * √2 + KM) / (KM + 24 * √2) = (24 * √2 – KM) / (KM – 7 * √2).

(KM + 24 * √2) * (24 * √2 – KM) = (7 * √2 + KM) * (KM – 7 * √2).

(24 * √2) ^ 2 – KM ^ 2 = KM ^ 2 – (7 * √2) 2.

2 * KM ^ 2 = 1152 + 98 = 1250.

KM = √ (1250/2) = √625 = 25 cm.

Answer: The length of the segment is 25 cm.