The diagonals of a trapezoid are 8 and 15. Find the area of this trapezoid if its midline is 8.5.

univerkov | May 30, 2021 | education

Let ABCD be a given trapezoid (AD and BC of the base), AC = 8, BD = 15, (BC + AD) = 8.5.

Since (BC + AD) = 8.5, then BC + AD = 17.

Let us draw CE parallel to BD (E belongs to the extension of side AD). The quadrilateral DBCE is a parallelogram (BC is parallel to AD and BD is parallel to CE), DE = BC, CE = BD.

Hence, AE = AD + DE = BC + AD = 17.

Let’s draw the height of CH (H belongs to AD).

The area of the trapezoid ABCD is equal to: (BC + AD) / 2 * CH = 17/2 * CH.

The area of the ACE triangle is equal to: AE / 2 * CH = 17/2 * CH.

Therefore, the areas of the trapezoid ABCD and the triangle ACE are equal.

Find the area of the ACE triangle using Heron’s formula: S = √ (p (p – a) (p – b) (p – c)) (p is the half perimeter of the triangle, a, b and c are the sides of the triangle).

AC = 8, CE = 15, AE = 17.

p = (8 + 15 + 17): 2 = 20.

S = √ (20 (20 – 8) (20 – 15) (20 – 17)) = √ (20 * 12 * 5 * 3) = √ (100 * 36) = 10 * 6 = 60.

Answer: the area of the trapezoid is 60 units ².

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.