Sides of a Triangle are in Arithmetic Progression

Math Problem: Sides of a Triangle are in arithmetic progression, then find the sines of angles

If the sides of the right-angle triangle are in arithmetic progression, then what are the sines of angles

Solution

Sides of the triangle are in arithmetic progression, so assume the sides of the triangles are x − a, x and x + a

Now apply pythegorus therem in triangle, then

AC² = AB² + BC²

⇒ (x + a)² = x² + (x − a)²

⇒ x² + 2ax + a² = x² + x² − 2ax + a²

⇒ x² = 4ax

⇒ x = 4a

From triangle

sin A = BC / AC = x / (x + a) = 4a / (4a + a)

⇒ sin A = 4/5

sin B = sin 90°

⇒ sin B = 1

sin C = AB / AC = (x − a) / (x + a) = (4a − a) / (4a + a)

⇒ sin C = 3/5