Two parallel lines, L_1 and L_2, are situated in a plane with L_1 above L_2. A regular pol… — JAMB UTME

Two parallel lines, $L_1$ and $L_2$ , are situated in a plane with $L_1$ above $L_2$ . A regular polygon of $n$ sides has one of its sides, $AB$ , lying entirely on line $L_1$ . An adjacent side, $BC$ , extends downwards into the region between $L_1$ and $L_2$ . A line segment $BD$ is drawn from vertex $B$ perpendicular to line $L_2$ , meeting $L_2$ at point $D$ . The segment $BD$ lies inside the interior angle $\angle ABC$ of the polygon and divides $\angle ABC$ into two angles, $\angle ABD$ and $\angle DBC$ , such that $\angle ABD : \angle DBC = 3 : 2$ . What is the total number of diagonals of this regular polygon?

Answer: 54

Answer

Because line L_1 is parallel to line L_2 and BD is perpendicular to L_2, BD is also perpendicular to L_1, making angle ABD equal to 90°. Based on the 3:2 ratio, angle DBC equals 60°, giving a total interior angle of 150°. Solving (n - 2) × 180° / n = 150° yields n = 12 sides. Substituting n = 12 into the diagonal formula n(n - 3) / 2 gives 54 diagonals.

Step-by-Step Solution

Determine the measure of angle ABD using parallel line properties

\angle ABD = 90^\circ

Because line L_1 is parallel to line L_2 and line segment BD is perpendicular to L_2, BD is also perpendicular to L_1.

Calculate the measure of angle DBC from the given angle ratio

\angle DBC = 60^\circ

With \angle ABD : \angle DBC = 3 : 2 and \angle ABD = 90^\circ, setting 3 ratio units equal to 90^\circ gives 1 unit = 30^\circ, so 2 units = 60^\circ.

Find the total interior angle measure of the regular polygon

\angle ABC = 150^\circ

The interior angle \angle ABC is the sum of \angle ABD and \angle DBC, which gives 90^\circ + 60^\circ = 150^\circ.

Determine the number of sides n of the regular polygon

n = 12

Using the regular polygon interior angle formula \frac{(n - 2) \times 180^\circ}{n} = 150^\circ, solving for n yields 180n - 360 = 150n \implies 30n = 360 \implies n = 12.

Calculate the total number of diagonals

Applying the polygon diagonal formula D = \frac{n(n - 3)}{2} with n = 12 gives D = \frac{12 \times (12 - 3)}{2} = \frac{12 \times 9}{2} = 54.

Key Concept

Combining parallel line perpendicularity properties, angle partitioning, regular polygon interior angle measure, and diagonal count calculation

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