I have three triangles drawn on a piece of paper. The first triangle has angles labeled A, B, and C. Angle A is a right angle, angle B is 45 degrees, and the side AC is labeled as 4. The second triangle also has angles labeled A, B, and C. Angle A is a right angle, angle B is 30 degrees, and the side BC is labeled as 8. The third triangle has angles labeled A, B, and C. Angle C is a right angle, angle B is 60 degrees, and the side BC is labeled as 7. Please analyze these triangles.

Triangle Analysis:

Here's an analysis of each triangle based on the provided information:

1. Triangle 1:
   • Angle A = 90° (Right Angle)
   • Angle B = 45°
   • Side AC = 4
   • Since the sum of angles in a triangle is 180°, Angle C = 180° - 90° - 45° = 45°.
   • This is a 45-45-90 triangle (an isosceles right triangle). Therefore, side AB = side AC = 4.
   • Using the Pythagorean theorem, BC² = AB² + AC² = 4² + 4² = 16 + 16 = 32. So, BC = √32 = 4√2.
2. Triangle 2:
   • Angle A = 90° (Right Angle)
   • Angle B = 30°
   • Side BC = 8
   • Since the sum of angles in a triangle is 180°, Angle C = 180° - 90° - 30° = 60°.
   • This is a 30-60-90 triangle. In a 30-60-90 triangle:
   • The side opposite the 30° angle (AC) is half the hypotenuse (BC). So, AC = BC / 2 = 8 / 2 = 4.
   • The side opposite the 60° angle (AB) is √3 times the side opposite the 30° angle. So, AB = AC * √3 = 4√3.
3. Triangle 3:
   • Angle C = 90° (Right Angle)
   • Angle B = 60°
   • Side BC = 7
   • Since the sum of angles in a triangle is 180°, Angle A = 180° - 90° - 60° = 30°.
   • This is a 30-60-90 triangle. In a 30-60-90 triangle:
   • The side opposite the 60° angle (BC) is √3 times the side opposite the 30° angle (AC). So, 7 = AC * √3, which means AC = 7 / √3 = (7√3) / 3.
   • The hypotenuse (AB) is twice the side opposite the 30° angle (AC). So, AB = 2 * AC = 2 * (7√3) / 3 = (14√3) / 3.