# In an isosceles trapezoid, a large base = 36 cm. Lateral side 25 cm. Diagonal 29 cm. Find the area

Using Heron’s theorem, we determine the area of the triangle ACD.

Sacd = √р * (р – АС) * (р – СD) * (р – АD), where р is a triangle semiperimeter.

P = (AC + CD + AD) / 2 = (29 + 25 + 36) / 2 = 45 cm.

Sasd = √45 * 16 * 20 * 9 = √129600 = 360 cm2.

Also Sasd = AD * CH / 2.

CH = 2 * Sacd / AD = 2 * 360/36 = 20 cm.

From the right-angled triangle СDН, we determine the length of the segment DH.

DH ^ 2 = CD ^ 2 – DH ^ 2 = 625 – 400 = 225.

DН = 15 cm.

Since the trapezoid is isosceles, then AK = DH = 15 cm.

Then the length of the segment KН = AD – AK – D = 36 – 15 – 15 = 6 cm.

Quadrangle ВСНK is a rectangle, then BC = KH = 6 cm.

Determine the area of the trapezoid.

Savsd = (ВС + АD) * СН / 2 = (6 + 36) * 20/2 = 420 cm2.

Answer: The area of the trapezoid is 420 cm2.