Find the each exterior angle of a polygon if total no. of diagonals are 90 and all angles are equal

Answer

To solve the problem of finding each exterior angle of a polygon given that the total number of diagonals is 90 and all angles are equal, we can follow these steps:

### Step 1: Use the formula for the number of diagonals in a polygon.
The formula for the number of diagonals \( D \) in a polygon with \( n \) sides is given by:
\[ D = \frac{n(n-3)}{2} \]
We know that \( D = 90 \). Therefore, we can set up the equation:
\[ 90 = \frac{n(n-3)}{2} \]

### Step 2: Solve for \( n \).
To eliminate the fraction, multiply both sides by 2:
\[ 180 = n(n-3) \]
This simplifies to:
\[ n^2 - 3n - 180 = 0 \]
Now, we can solve this quadratic equation using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -3, c = -180 \).

### Step 3: Calculate the discriminant.
First, calculate the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot (-180) = 9 + 720 = 729 \]

### Step 4: Solve for \( n \).
Now substitute back into the quadratic formula:
\[ n = \frac{3 \pm \sqrt{729}}{2} \]
Since \( \sqrt{729} = 27 \), we have:
\[ n = \frac{3 \pm 27}{2} \]
Calculating the two possible values for \( n \):
1. \( n = \frac{30}{2} = 15 \)
2. \( n = \frac{-24}{2} = -12 \) (not valid since \( n \) must be positive)

Thus, \( n = 15 \).

### Step 5: Find each exterior angle.
The formula for each exterior angle \( E \) of a regular polygon is:
\[ E = \frac{360^\circ}{n} \]
Substituting \( n = 15 \):
\[ E = \frac{360^\circ}{15} = 24^\circ \]

### Final Answer:
Each exterior angle of the polygon is \( 24^\circ \).

---