In triangle ABC AB: AC = 3√2: 7, angle BAC = 45 degrees. Find the AC side if BC = 30 cm.

Let’s draw the height BH of the triangle ABC.

Let the side length AB = 3 * √2 * X cm, then AC = 7 * X cm.

The right-angled triangle ABH is isosceles, since its angle BAH = 45.

Then AH = BH, and Cos45 = AH / AB.

AH = AB * Cos45 = 3 * √2 * X * √2 / 2 = 3 * X cm.

AH = BH = 3 * X cm.

Then the length of the segment CH = AC – AH = 7 * X – 3 * X = 4 * X cm.

In a right-angled triangle BCH, according to the Pythagorean theorem, BC ^ 2 = BH ^ 2 + CH ^ 2.

90 = 9 * X ^ 2 + 16 * X ^ 2 = 25 * X ^ 2.

X ^ 2 = 900/25 = 36.

X = 6.

Then AC = 7 * 6 = 42 cm.

Answer: The length of the speaker side is 42 cm.