The ratio of the number of sides of two polygons is `1:2`, and the ratio of the sum of their angles is `3:8`. Find the number of sides in each polygon.

Options

A. `7` and `19`
B. `5` and `19`
C. `5` and `18`
D. `8` and `11`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let the number of sides of the first polygon be \( x \) and the number of sides of the second polygon be \( 2x \) since the ratio of the number of sides is \( 1:2 \).

### Step 2: Write the Formula for the Sum of Angles
The sum of the interior angles of a polygon with \( n \) sides is given by the formula:
\[ \text{Sum of angles} = (n - 2) \times 180^\circ \]
Thus, for the first polygon (with \( x \) sides):
\[ S_1 = (x - 2) \times 180^\circ \]
For the second polygon (with \( 2x \) sides):
\[ S_2 = (2x - 2) \times 180^\circ \]

### Step 3: Set Up the Ratio of the Sum of Angles
According to the problem, the ratio of the sum of their angles is \( 3:8 \). Therefore, we can write:
\[ \frac{S_1}{S_2} = \frac{3}{8} \]
Substituting the expressions for \( S_1 \) and \( S_2 \):
\[ \frac{(x - 2) \times 180^\circ}{(2x - 2) \times 180^\circ} = \frac{3}{8} \]
We can cancel \( 180^\circ \) from both sides:
\[ \frac{x - 2}{2x - 2} = \frac{3}{8} \]

### Step 4: Cross-Multiply to Solve for \( x \)
Cross-multiplying gives:
\[ 8(x - 2) = 3(2x - 2) \]
Expanding both sides:
\[ 8x - 16 = 6x - 6 \]

### Step 5: Rearrange the Equation
Now, rearranging the equation to isolate \( x \):
\[ 8x - 6x = -6 + 16 \]
This simplifies to:
\[ 2x = 10 \]

### Step 6: Solve for \( x \)
Dividing both sides by 2:
\[ x = 5 \]

### Step 7: Find the Number of Sides in Each Polygon
Now we can find the number of sides in both polygons:
- First polygon: \( x = 5 \)
- Second polygon: \( 2x = 2 \times 5 = 10 \)

### Final Answer
The number of sides in the first polygon is **5**, and the number of sides in the second polygon is **10**.

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