# The angle at the apex opposite to the base of the isosceles triangle is 150 and the side of the triangle is 7. Find the area of this triangle.

A triangle is three points that do not lie on one straight line, connected by segments. In this case, the points are called the vertices of the triangle, and the segments are called its sides.

An isosceles triangle is a triangle in which the sides are equal.

In order to find the area of ​​a triangle, you need to multiply half of its base by the height:

S = AC / 2 h, where;

S is the area of ​​the triangle;

AC – base;

h – height.

To do this, you need to calculate the length of the height VN and half of the base (segment AH).

The height of an isosceles triangle, lowered to its base, is the bisector of the angle opposite to the base.

In this way:

∠АВН = ∠НВС = ∠АВС / 2;

∠АВН = ∠НВС = 150º / 2 = 75º.

Consider the triangle ΔАВН.

To calculate the ВН, we apply the cosine theorem. The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:

cos B = BH / AB;

BH = AB · cos B;

cos 75º ≈ 0.2588;

BH = 7 0.2588 ≈ 1.8 cm.

To calculate AН, we use the Pythagorean theorem, according to which the square of the hypotenuse is equal to the sum of the squares of the legs:

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

AH ^ 2 = AB ^ 2 – BH ^ 2;

AH ^ 2 = 7 ^ 2 – 1.8 ^ 2 = 49 – 3.24 = 45.76;

AH = √45.76 ≈ 6.8 cm;

S = AH · h;

S = 1.8 6.8 = 12.24 cm2

Answer: the area of ​​the triangle is 12.24 cm2.

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