Mastering Triangles: Sines, Cosines, and Area Calculations

A triangle has angles measuring 30 degrees and 60 degrees. If the side opposite the 30-degree angle is 5 cm, what is the length of the side opposite the 60-degree angle using the law of sines?

• 5 cm

• 10 cm

• 5√3 cm

• 3√3 cm

In a right triangle, one angle measures 45 degrees and the adjacent side is 10 m. What is the length of the opposite side using trigonometric ratios?

• 15 m

• 10 m

• 20 m

• 5 m

A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Identify the type of triangle and calculate its area using Heron's formula.

• The triangle is an isosceles triangle and its area is 48 cm².

• The triangle is a right triangle and its area is 84 cm².

• The triangle is a scalene triangle and its area is 100 cm².

• The triangle is an equilateral triangle and its area is 42 cm².

A boat sails 10 km north and then 6 km east. What is the distance from the starting point to the boat's final position? Use the law of cosines to find the answer.

• 11.66 km

• 15.00 km

• 12.00 km

• 8.00 km

In triangle ABC, angle A measures 50 degrees, angle B measures 70 degrees, and side a is 12 m. Calculate the length of side b using the law of sines.

• 14.97 m

• 16.25 m

• 10.50 m

• 12.75 m

A ladder leans against a wall, forming a 60-degree angle with the ground. If the foot of the ladder is 4 m from the wall, how long is the ladder?

• 8 m

• 12 m

• 10 m

• 6 m

A triangle has sides of lengths 8 cm, 15 cm, and 17 cm. Determine if it is a right triangle and find its area.

• No, it is not a right triangle. The area is 68 cm².

• Yes, it is a right triangle. The area is 80 cm².

• Yes, it is a right triangle. The area is 60 cm².

• No, it is a right triangle. The area is 30 cm².

A surveyor measures a triangle with sides of lengths 15 m, 20 m, and an included angle of 60 degrees. Calculate the area of the triangle using the formula A = 1/2 * a * b * sin(C).

• 80.50 m²

• 64.95 m²

• 75.00 m²

• 50.25 m²

A triangular park has sides measuring 30 m, 40 m, and 50 m. Identify the type of triangle and calculate its area using the law of cosines.

• 700 m²

• 600 m²

• 450 m²

• 500 m²

A kite is flying at an angle of elevation of 30 degrees from a point 50 m away. How high is the kite above the ground? Use trigonometric ratios to find the height.

• 15 m

• 40 m

• 50 m

• 28.87 m