The height CH is drawn in triangle CDE. It is known that angle D = 30 degrees, side DC = 18 cm

The height CH is drawn in triangle CDE. It is known that angle D = 30 degrees, side DC = 18 cm, angle E = 45 degrees. Find HE.

Consider a right-angled triangle СНD, in which the angle СНD = 90, since СН is the height to the side DE, and the angle СDE = 30 by condition.

Then the side of the CH lies against an angle of 30 degrees and, accordingly, is equal to half the length of the hypotenuse.

CH = CD / 2 = 18/2 = 9 cm.

In the triangle CHE, the angle CHE = 90, since CH is the height of the triangle CDE, and the angle HEC = 45 by condition. Then the angle HCE = 180 – 90 – 45 = 45.

The angles at the base of the CE of the CHE triangle are equal, therefore, the CHE triangle is isosceles, the side CH = HE = 9 cm.

Answer: Section HЕ = 9 cm.