Question

The ratio of areas of a regular pentagon to the polygon formed by joining the points of intersection of its diagonals is (a) \(1: 4\) (b) \(1: 16 \sin ^{2} 14^{\circ}\) (c) \(1: 16 \sin ^{4} 18^{\circ}\) (d) \(4: \sin ^{4} 18^{\circ}\)

Short answer

Answer: \(1: 16 \sin ^{4} 18^{\circ}\)

Step-by-step solution

Draw a diagram and label the points of intersection

Draw a regular pentagon and label its vertices A, B, C, D, and E. Draw all the diagonals of the pentagon by connecting non-adjacent vertices. Now we can label the points of intersection of these diagonals as P, Q, R, S, and T, where P is the intersection of AD and BC, Q is the intersection of BE and AC, and so on.

Identify important angles

We can see that angle A is \(\frac{3 \times 180^{\circ}}{5} = 108^{\circ}\). Also notable is angle DAC which is \(\frac{180^{\circ}}{5} = 36^{\circ}\). To help us compute the area of various parts of the figure, we need to find the internal angles of triangle \(ABC\). Since the triangle is isosceles, angle BAC = angle BCA = \(18^{\circ}\).

Compute the area of triangle ABC

Let the side length of the regular pentagon be a. Then we can use base length BC = a and height, which is equal to \(a\sin 18^{\circ}\), to compute the area of triangle ABC. Area of ABC = \(\frac{1}{2} \times a \times a\sin 18^{\circ}\)

Compute the area of triangle APQ

First, we find the length of AP. With AP = x, we note that \(\frac{x}{a} = \frac{\sin 18^{\circ}}{\sin 54^{\circ}}\) or \(x = a\frac{\sin 18^{\circ}}{\sin 54^{\circ}}\). Now we can compute the area of triangle APQ. Area of APQ = \(\frac{1}{2} \times x \times x\sin 18^{\circ}\)

Compute the area of the pentagon formed by diagonals

There are five equal isosceles triangles APQ, BPE, CRF, DSQ, and ESX that form the interior pentagon. The area of this pentagon is equal to the sum of the areas of these triangles. Area of inner pentagon = 5 \(\times\) Area of APQ = \(\frac{5}{2}\times a^2\frac{\sin^2 18^{\circ}}{\sin^2 54^{\circ}}\sin 18^{\circ}\)

Compute the area ratio of the regular pentagon to the inner pentagon

Now that we have the area of the pentagon formed by connecting the points of intersection of its diagonals and area of outer pentagon, we can compute the desired ratio. Area ratio = \(\frac{Area \space out\ part}{Area \space inner \space pentagon} = \frac{\frac{a^2\sin 18^{\circ}-\frac{5}{2}\times a^2\frac{\sin^2 18^{\circ}}{\sin^2 54^{\circ}}\sin 18^{\circ}}}{\frac{5}{2}\times a^2\frac{\sin^2 18^{\circ}}{\sin^2 54^{\circ}}\sin 18^{\circ}}\) After simplifying, we get the area ratio: Area ratio = \(1: 16 \sin^2 18^{\circ}\) Thus, the correct answer is option (c) \(1: 16 \sin ^{4} 18^{\circ}\).

Key concepts

Understanding a Regular Pentagon

A regular pentagon is a five-sided polygon where all sides are of equal length and all interior angles are equal. Each angle in a regular pentagon is calculated using the formula:

• Angle = \( \frac{(5-2) \times 180^{\circ}}{5} = 108^{\circ} \)

This means that each angle in a regular pentagon measures 108°. A regular pentagon has unique and symmetrical properties, making it a fascinating geometrical figure.To construct a regular pentagon:

• Start by drawing one side of a specified length.
• Using a compass, swing arcs from each endpoint, creating intersections that form the remaining vertices.
• Connect these vertices to complete the pentagon.

Regular pentagons are often used in tiling patterns and are found in nature, such as in the shape of flowers and other biological structures.

Exploring Diagonals Intersection in a Pentagon

In a pentagon, diagonals are lines connecting non-adjacent vertices. For a regular pentagon, there are five diagonals. When these diagonals intersect, they form a smaller internal pentagon, creating a pattern of star-shaped lines.The points where these diagonals intersect create new vertices that form another, smaller pentagon. To understand the intersection:

• Draw all possible diagonals connecting non-adjacent vertices.
• Label intersections as \( P, Q, R, S, \) and \( T \).

This formation is important for understanding geometric properties such as symmetry and internal ratios.Such intersections are crucial in determining various geometric properties, especially when calculating areas of different sections formed by these intersections.

Isosceles Triangles in a Pentagon

Isosceles triangles are those with two sides of equal length and consequently two equal angles. Within a regular pentagon, drawing diagonals creates multiple isosceles triangles. For instance, examining triangle \( ABC \) can help understand its properties:

• Since it's isosceles, if one angle is known, the other two equal angles can easily be determined.
• From the problem, angle \( BAC = 18^{\circ} \), hence angle \( BCA \) is also \( 18^{\circ} \).

Calculating areas of these triangles helps in understanding further geometrical properties of the pentagon. By knowing the base and height, we can calculate the area of these triangles with:\[ Area = \frac{1}{2} \times base \times height \]This becomes essential in more complex calculations, such as finding the area of the inner pentagon.

Calculating Areas within the Pentagonal Structure

Area calculation within a regular pentagon, especially when considering the smaller inner pentagon formed by diagonals, involves several mathematical steps. The process involves:

• Calculating the area of the regular outer pentagon using known formulas.
• Deducting the area of the smaller pentagon formed by diagonal intersections.

When calculating the area of the larger pentagon, use trigonometry and known side lengths. The key to understanding the ratio of different pentagonal areas lies in computing areas of the smaller isosceles triangles, as shown in the original problem:\[ Area = \frac{1}{2} \times a \times a \sin 18^{\circ} \]The complexity comes from finding exact points of diagonal intersections and employing trigonometric identities to solve the area ratios.The intriguing result: The area ratio shows how much larger the outer regular pentagon is in comparison to the smaller one formed inside.