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Cosine Rule Problems

Problem 1 In the ABC we have AB = 5cm, AC = 6cm, α = 60°. Find the length of the side BC.
Answer: √31cm.

Problem 2 Define the type of of the triangle by it's angle, if the lengths of it's sides are:
a) 13, 14, 15;
b) 12, 35, 37;
c) 13, 15, 24.
Answer:a) - obtuse-angled triangle;
b) - right-angled triangle;
c) - acute-angled triangle.

Problem 3 Find the size of the angle γ of the ABC, if:
a) a = 2√3cm, b = 3cm, c = √3cm;
b) a = 11cm, b = 60cm, c = 61cm.
Answer:a) - 30°;
b) - 90°.

Problem 4 In the ABC we have AC = 3cm, BC = √5cm, α = 45°. Find the length of the side AB.
Answer:√2cm or 2√2cm.

Problem 5 The length of the diagonale of the rectangular is 32cm, and the angle between the diagonales is 135°. Find the lengths of the sides of the rectangular.
Answer:16√2 - √2cm and 16√2 + √2cm.

Problem 6 The center of the circumference, inscribed in a right-angled triangle, is at distance √5 and √10 from the two ends of the hypotenuse. Find the length of the hypotenuse.
Answer: 5cm.

Problem 7 The center of the inscribed in the ABC circumference is at distance 7 and 3√3 from the points A and B. Find the length of the side AB, if the angle at the point C is 120°.
Answer: √139cm.