The angles of a quadrilateral are in AP, and the greatest angle is $3$ times the smallest. Find all four angles.
Step-by-Step Solution
Key Concept: Write the four angles as (a-3d),(a-d),(a+d),(a+3d) so their sum is automatically 4a; use the angle-sum property and the given ratio condition.
Let the angles be $a-3d, a-d, a+d, a+3d$ (in AP with common difference $2d$). Sum $=4a=360^\circ\Rightarrow a=90^\circ$. [1.0 Mark]
Given greatest $=3\times$ smallest: $a+3d=3(a-3d)\Rightarrow a+3d=3a-9d\Rightarrow12d=2a\Rightarrow d=\dfrac{a}{6}=15^\circ$. [1.0 Mark]
The angles are $90-45=45^\circ,\ 90-15=75^\circ,\ 90+15=105^\circ,\ 90+45=135^\circ$. [1.0 Mark]
Correct Answer: