In an isosceles trapezoid, a smaller base is 5 cm, the sides are 4 cm slopes with a base equal

In an isosceles trapezoid, a smaller base is 5 cm, the sides are 4 cm slopes with a base equal to 60 degrees. Find the area of the trapezoid?

An isosceles trapezoid is called, in which the sides are equal and the angles at the bases are equal. Thus:

AB = CD;

∠А = ∠D;

∠В = ∠С.

The area of ​​the trapezoid is equal to the product of the half-sum of its bases by the height:

S = (a + b) / 2 h, where:

S is the area of ​​the trapezoid;

a – smaller base;

b – larger base;

h – height.

To do this, you need to find the length of the larger base and the height of the trapezoid.

Let’s draw the heights of the VN and SK. Since the trapezoid is isosceles, the segments AH and KD are equal.

Since the segment of the larger base located between the heights is equal to the length of the smaller base, then:

NK = BC;

AD = BC + AH + KD.

To calculate the segments AH and KD, we apply the cosine theorem:

cos A = AH / AB;

AH = AB · cos A;

cos 60º = 1/2;

AH = 4 1/2 = 4/2 = 2 cm;

KD = AH = 2 cm;

AD = 5 + 2 + 2 = 9 cm.

To calculate the height of the VN, we apply the Pythagorean theorem:

AB ^ 2 = BH ^ 2 + AH ^ 2;

BH ^ 2 = AB ^ 2 – AH ^ 2;

BH ^ 2 = 4 ^ 2 – 2 ^ 2 = 16 – 4 = 12;

BH = √12 = 3.5 cm.

S = (5 + 9) / 2 * 3.5 = 14/2 * 3.5 = 24.5 cm2.

Answer: the area of ​​the trapezoid is 24.5 cm2.