Question

Regular polygons A and B have number of sides in the ratio \(1:2\) and interior angles in the ratio \(3:4\).Then the number of sides of B equals

Answer 1

Correct Answer: 10

The problem asks us to determine the number of sides of two regular polygons A and B, given the ratio of their interior angles is 3:4. Let's break it down step by step: 

Step 1: Defining Variables

Let n be the number of sides of polygon A, and the number of sides of polygon B will be 2n, since the ratio of the number of sides is 1:2.

Step 2: Formula for Interior Angles

The sum of the interior angles of a polygon with n sides is given by the formula:

\(\text{Sum of interior angles} = (n - 2) \times 180^\circ\)

The measure of each interior angle in a regular polygon is:

\(\text{Each interior angle} = \frac{(n - 2) \times 180}{n}\)

Step 3: Setting up the Ratio of Interior Angles

We are given that the ratio of the interior angles of polygons A and B is 3:4. Hence, we can set up the following ratio:

\(\frac{ \frac{(n - 2) \times 180}{n} }{ \frac{(2n - 2) \times 180}{2n} } = \frac{3}{4}\)

Step 4: Simplifying the Equation

Now, we simplify the above equation:

\(\frac{ (n - 2) \times 180 }{ n } \times \frac{ 2n }{ (2n - 2) \times 180 } = \frac{3}{4}\)

This simplifies to:

\(\frac{ (n - 2) \times 2n }{ n \times (2n - 2) } = \frac{3}{4}\)

Canceling out the common factors and simplifying further:

\(\frac{ 2(n - 2) }{ 2n - 2 } = \frac{3}{4}\)

Step 5: Solving for n

We now solve for n by cross-multiplying:

\(4 \times 2(n - 2) = 3 \times (2n - 2)\)

Expanding both sides:

\(8(n - 2) = 6n - 6\)

Now, simplifying:

\(8n - 16 = 6n - 6\)

Move all terms involving n to one side:

\(8n - 6n = -6 + 16\)

\(2n = 10\)

Dividing by 2:

\(n = 5\)

Step 6: Finding the Number of Sides for Polygon B

Since polygon B has 2n sides, we have:

\(2 \times 5 = 10\)

Conclusion

Thus, polygon A has 5 sides and polygon B has 10 sides.

Answer 2

Let the number of sides of polygons A and B be \(n\) and \(2n\), respectively.

The sum of the interior angles of a polygon is given by the formula:

\((n - 2) \times 180^\circ\)

For a regular polygon, each interior angle is given by:

\(\frac{(n - 2) \times 180^\circ}{n}\)

Now, we are given that the ratio of the interior angles of polygons A and B is 3:4. Hence, we can write:

\(\frac{\frac{(n - 2) \times 180^\circ}{n}}{\frac{(2n - 2) \times 180^\circ}{2n}} = \frac{3}{4}\)

We simplify this equation:

\(\frac{\frac{(n - 2) \times 180^\circ}{n}}{\frac{2(n - 1) \times 180^\circ}{2n}} = \frac{3}{4}\)

This simplifies further to:

\(\frac{n - 2}{n - 1} = \frac{3}{4}\)

Now, solving for \(n\):

\(4(n - 2) = 3(n - 1)\)

\(4n - 8 = 3n - 3\)

\(n = 5\)

Therefore, polygon A has 5 sides, and polygon B has \(2 \times 5 = 10\) sides.

Conclusion

Thus, the answer is 10.