The acute angle of a parallelogram is 60 degrees, the smaller diagonal is inclined to the larger side at an angle of 30 degrees, find the area of the parallelogram if its larger side is 20
A parallelogram is a quadrilateral in which opposite sides are pairwise parallel.
To calculate the area of a parallelogram, we use the area formula in terms of sides and angles:
S = a b sin α = a b sin ß, where:
S is the area of the parallelogram;
a – sides AB or SD;
b – side of the aircraft or air pressure;
α is the angle between AB and AD;
ß is the angle between AB and BC.
In order to find the area of the parallelogram, we calculate the length of the side AB.
To do this, consider the triangle ΔABD.
Since the sum of all the angles of the triangle is 180º, then:
∠BAD + ∠ABD + ∠BDA = 180º;
∠ABD = 180º – ∠BAD – ∠BDA;
∠BDA = 30º;
∠BAD = 60º;
∠ABD = 180º – 60º – 30º = 90º.
Since the angle ∠ABD is right, this triangle is right-angled.
To calculate AB, we apply the cosine theorem. The cosine of an acute angle of a right-angled triangle is the ratio of the adjacent leg to the hypotenuse:
cos A = AB / AD;
AB = AD ∙ cos A;
cos 60º = 1/2;
AB = 20 1/2 = 20/2 = 10 cm;
sin 60º = sin 120º ≈ 0.866;
S = 20 10 ∙ 0.866 = 173.2 cm2.
Answer: the area of the parallelogram is 173.2 cm2.