In a triangle abc: cos C = -0.8, BC = 25, AB = 39. Find the area of a triangle

| September 19, 2021 | education

Let us write down the cosine theorem for the side AB of the triangle ABC.

AB ^ 2 = AC ^ 2 + BC ^ 2 – 2 * AC * BC * CosC.

39 ^ 2 = AC ^ 2 + 25 ^ 2 – 2 * AC * 25 * (-0.8).

1521 = AC ^ 2 + 625 + 40 * AC.

AC ^ 2 + 40 * AC – 896 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = 40 ^ 2 – 4 * 1 * (-896) = 1600 + 3584 = 5184.

X1 = (-40 – √5184) / (2 * 1) = (-40 – 72) / 2 = -112 / 2 = -56. Doesn’t match because <0.

X2 = (-40 + √5184) / (2 * 1) = (-40 + 72) / 2 = 32/2 = 16.

AC = 16 cm.

Let us determine the area of ​​a triangle by Heron’s theorem.

S = √p * (p – AB) * (p – BC) * (p – AC), where p is the semiperimeter of the triangle.

p = (AB + BC + AC) / 2 = (39 + 25 + 16) / 2 = 40 cm.

S = √40 * (40 – 39) * (40 – 25) * (40 – 16) = √40 * 1 * 15 * 24 = √14400 = 120 cm2.

Answer: The area of ​​the triangle is 120 cm2.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.