Two regular polygons, P_1 and P_2, have n sides and 2n sides respectively. If the sum of o… — JAMB UTME

Two regular polygons, $P_1$ and $P_2$ , have $n$ sides and $2n$ sides respectively. If the sum of one interior angle of $P_1$ and one exterior angle of $P_2$ is $150^\circ$ , what is the total number of diagonals of polygon $P_2$ ?

A
9
54Answer
C
66
D
108

Answer

The total number of diagonals of polygon

P_2

is 54.

The interior angle of an

n

-sided polygon is

180^\circ - \frac{360^\circ}{n}

and the exterior angle of a

2n

-sided polygon is

\frac{180^\circ}{n}

. Adding these gives

180^\circ - \frac{180^\circ}{n} = 150^\circ

, which yields

n = 6

. Polygon

P_2

therefore has

12

sides, and the number of diagonals is

\frac{12 \times 9}{2} = 54

Step-by-Step Solution

Express the interior angle of

P_1

and the exterior angle of

P_2

in terms of

n

Interior angle of

P_1 = 180^\circ - \frac{360^\circ}{n}

; Exterior angle of

P_2 = \frac{360^\circ}{2n} = \frac{180^\circ}{n}

The interior angle of an

n

-sided regular polygon is

180^\circ - \frac{360^\circ}{n}

, and the exterior angle of a

2n

-sided regular polygon is

\frac{360^\circ}{2n}

Set up and solve the angle sum equation.

\left(180^\circ - \frac{360^\circ}{n}\right) + \frac{180^\circ}{n} = 150^\circ \implies 180^\circ - \frac{180^\circ}{n} = 150^\circ \implies \frac{180^\circ}{n} = 30^\circ \implies n = 6$.

The sum of the two angles is given as

150^\circ

Determine the number of sides of polygon

P_2

Polygon

P_2

has

2n = 2(6) = 12

sides.

Polygon

P_2

has

2n

sides.

Calculate the total number of diagonals for a polygon with 12 sides using

D = \frac{k(k-3)}{2}

D = \frac{12(12 - 3)}{2} = \frac{12 \times 9}{2} = 54.

The formula for the number of diagonals in a polygon with

k

sides is

\frac{k(k-3)}{2}

Key Concept

Interior and exterior angles of regular polygons and the polygon diagonal count formula

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic