The angles of a triangle are in AP; the greatest is twice the least. Find all angles.
40°, 60°, 80°
Step 1: In an Arithmetic Progression (AP), three terms can be written as:
\(a - d, \, a, \, a + d\)
Here, \(a\) is the middle angle, and \(d\) is the common difference.
Step 2: It is given that the greatest angle is twice the least angle.
So, \(a + d = 2(a - d)\).
Step 3: Simplify:
\(a + d = 2a - 2d\)
\(a - 3d = 0 \Rightarrow a = 3d\)
Step 4: The sum of angles in a triangle is \(180^{\circ}\).
\((a - d) + a + (a + d) = 180^{\circ}\)
That is, \(3a = 180^{\circ}\).
Step 5: Solve for \(a\):
\(a = \dfrac{180^{\circ}}{3} = 60^{\circ}\).
Step 6: Since \(a = 3d\),
\(60^{\circ} = 3d \Rightarrow d = 20^{\circ}\).
Step 7: Now find the three angles:
Final Answer: The angles are \(40^{\circ}, 60^{\circ}, 80^{\circ}\).