Question

The sum of an interior angle of rectangular polygon is 3x° while the exterior angle is(x-20)° . Find the numbers of sides of the polygon

Answer 1

Let the number of sides of the polygon be \( n \).

The sum of the interior angles of an \( n \)-sided polygon is given by the formula:

\[
\text{Sum of interior angles} = (n - 2) \times 180°
\]

According to the problem, this sum is equal to \( 3x° \):

\[
(n - 2) \times 180 = 3x
\]

The exterior angle of a polygon is related to the number of sides by the formula:

\[
\text{Exterior angle} = \frac{360°}{n}
\]

The problem states that the exterior angle is \( (x - 20)° \):

\[
\frac{360}{n} = x - 20
\]

We now have two equations:

1. \((n - 2) \times 180 = 3x\)
2. \(\frac{360}{n} = x - 20\)

**Step 1: Solve for \( x \) from the second equation:**

\[
x = \frac{360}{n} + 20
\]

**Step 2: Substitute this expression for \( x \) into the first equation:**

Substituting \( x \) into the first equation gives:

\[
(n - 2) \times 180 = 3\left(\frac{360}{n} + 20\right)
\]

Expanding both sides:

\[
180n - 360 = \frac{1080}{n} + 60
\]

**Step 3: Eliminate the fraction by multiplying through by \( n \):**

\[
n(180n - 360) = 1080 + 60n
\]

This simplifies to:

\[
180n^2 - 360n = 1080 + 60n
\]

**Step 4: Rearrange the equation:**

Combining all terms gives:

\[
180n^2 - 360n - 60n - 1080 = 0
\]
\[
180n^2 - 420n - 1080 = 0
\]

**Step 5: Divide through by 60 to simplify:**

\[
3n^2 - 7n - 18 = 0
\]

**Step 6: Factor the quadratic equation:**

To factor \( 3n^2 - 7n - 18 \), we can look for two numbers that multiply to \( 3 \cdot -18 = -54 \) and add up to \( -7 \). The numbers \( -9 \) and \( 2 \) work:

\[
3n^2 - 9n + 2n - 18 = 0
\]
\[
3n(n - 3) + 6(n - 3) = 0
\]
\[
(n - 3)(3n + 6) = 0
\]

**Step 7: Solve for \( n \):**

Setting each factor to zero gives:

1. \( n - 3 = 0 \) → \( n = 3 \)
2. \( 3n + 6 = 0 \) → \( n = -2 \) (not valid for a polygon)

**Final Answer:**

The number of sides of the polygon is

\[
\boxed{3}
\]

Verifying, a triangle does indeed have 3 sides, where:

- Interior angle sum = \( 3x = (3 - 2) \times 180 = 180° \)
- Exterior angle = \( \frac{360}{3} = 120° = (x - 20) \) gives \( x = 140°\), confirming the problem's conditions.