# In a triangle ABC ∠ACB = 90 °, AB = 10, cos∠ABC = 0.6. Find the area of triangle ABC.

A triangle in which one of the angles is 90º is called rectangular.

To calculate the area of ​​a given triangle, it is most convenient to use Heron’s formula:

S = √p (p – a) (p – b) (p – c); where:

S is the area of ​​the triangle;

p – semi-perimeter (p = (a + b + c) / 2);

a – side AB;

b – aircraft side;

c – speaker side.

To do this, it is tedious to find the length of the unknown sides of the BC and AC.

To calculate the side of the BC, 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 B = BC / AB;

BC = AB · cos B;

BC = 10 ∙ 0.6 = 6 cm.

To calculate the AC side, we apply the Pythagorean theorem, according to which the square of the hypotenuse is equal to the sum of the squares of the legs:

AB ^ 2 = BC ^ 2 + AC ^ 2;

AC ^ 2 = AB ^ 2 – BC ^ 2;

AC ^ 2 = 10 ^ 2 – 6 ^ 2 = 100 – 36 = 64;

AC = √64 = 8 cm.

p = (10 + 6 + 8) / 2 = 12 cm;

S = √ (12 ∙ (12 – 10) ∙ (12 – 6) ∙ (12 – 8)) = √ (12 ∙ 2 ∙ 6 ∙ 4) = √576 = 24 cm2.

Answer: the area of ​​the triangle is 24 cm2.

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