In a triangle ABC AB = 5 cm, BC = 4 cm, and its area is 5 roots of 3 cm2. find the third side of the triangle

In a triangle ABC AB = 5 cm, BC = 4 cm, and its area is 5 roots of 3 cm2. find the third side of the triangle if you know that angle B is acute

We calculate the sine of the angle between these sides through the area of the triangle and the length of its two sides.

Saavs = AB * BC * SinABC / 2.

5 * √3 = 5 * 4 * SinABC/ 2.

SinABC = 10 * √3 / 20 = √3 / 2.

Then Cos2ABC = 1 – Sin2ABC = 1 – 3/4 = 1/4.

CosABC = 1/2.

Angle ABC = 600.

AC ^ 2 = BC ^ 2 + AB ^ 2 – 2 * BC * AB * Cos60 = 16 + 25 – 2 * 4 * 5 * 1/2 = 21.

AC = √21 cm.

Answer: The length of the third side is √21 cm.

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