Side BC of triangle ABC is 20 cm. Angle A is 45 degrees. Angle C is 60 degrees. Calculate the length of the CA side.

Let’s build from the top B the height of the ВН. Then the triangles ВСНand AВН are rectangular.

In a right-angled triangle СВН, the angle СВН = (180 – 90 – 60) = 30.

Then the side of the CH lies against the angle of 300, and therefore is equal to half the length of the hypotenuse BC.

CH = BC / 2 = 20/2 = 10 cm.

By the Pythagorean theorem, we determine the length of the leg BN.

BH ^ 2 = BC ^ 2 – CH ^ 2 = 400 – 100 = 30.

BH = 10 * √3 cm.

In a right-angled triangle ABН, the angle BAН = 45, then this triangle is isosceles, AH = BH = 10 * √3 cm.

Then AC = CH + AH = 10 + 10 * √3 = 10 * (1 + √3) cm.

Answer: The length of the speaker side is 10 * (1 + √3) cm.