The bases BC and AD of the trapezoid ABCD are equal to 3 and 6, the diagonals intersect at point O

| May 10, 2021 | education

The bases BC and AD of the trapezoid ABCD are equal to 3 and 6, the diagonals intersect at point O, the sum of the areas of the triangles AOB and COD is 40. Find the height of the trapezoid.

Let us construct, through the point O, the height of the NM.

Let the height HM = X cm, and the segment OH = Y cm.

Triangles BOC and AOD are similar in two angles with a similarity coefficient K = BC / AD = 1/2.

Then OH / OM = Y / (X – Y) = 1/2.

2 * Y = X – Y.

X = 3 * Y.

The area of ​​the triangle BОС is equal to: Sвос = У * ВС / 2 = У * 3/2.

The area of ​​the triangle AOD is equal to: Svos = (X – Y) * AD / 2 = 3 * (X – Y).

The area of ​​the trapezium ABCD is equal to: Savsd = (BC + AD) * X / 2 = 9 * X / 2.

Savs = Svos + Saod + Saov + Ssod = Svos + Saod + 40.

9 * X / 2 = Y * 3/2 + 3 * (X – Y) + 40.

9 * X / 2 = (X / 3) * 3/2 + 3 * (X – X / 3) + 40.

– (X / 2) – 2 * X + 9 * X / 2 = 40.

9 * X – 5 * X = 80.

4 * X = 80.

NM = X = 80/4 = 20 cm.

Answer: The height of the trapezoid is 20 cm.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.