In trapezoid ABCD, angles A and B are straight. diagonal AC is the bisector of angle A and is equal to 6 cm.

| January 16, 2021 | education

In trapezoid ABCD, angles A and B are straight. diagonal AC is the bisector of angle A and is equal to 6 cm. Find the area of the trapezoid if the angle CDA is 60 degrees.

From the top of the trapezoid, we lower the height CH.

Since, by condition, AC is the bisector of a right angle, the angles ACН and СAН are equal to 450, and therefore the triangle ACН is isosceles and AH = CH.

Then, by the Pythagorean theorem, AC ^ 2 = AH ^ 2 + CH ^ 2 = 2 * AH ^ 2.

36 = 2 * AH ^ 2.

AH ^ 2 = 18.

AH = CH = BC = AB = 3 * √2 cm.

Consider a right-angled triangle СНD, in which the angle CDH = 60, and the leg СН = 3 * √2.

DH = CH / tan 60 = 3 * √2 / √3 = 3 * √2 * √3 / 3 = √6.

Then АD = AH * DH = 3 * √2 + √6

The area of ​​the trapezoid is equal to the sum of the area of ​​the square ABCH and the triangle CHD.

Sabch = AH ^ 2 = (3 * √2) ^ 2 = 18 cm2.

Schd = HD * CH / 2 = √6 * 3 * √2 / 2 = √3 * √2 * 3 * √2 / 2 = 3 * √3 cm2.

S = 18 + 3 * √3 cm2.

Answer: S = 18 + 3 * √3 cm2.



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.