Question

Construct a regular pentagon in a circle of radius \(4 \mathrm{~cm}\). The following sentences are the steps involved in the following constructions. Arrange them in sequential order from first to last. (a) With \(\mathrm{D}\) as centre and \(\mathrm{AB}\) as radius draw an arc which cuts the circle at the point \(\mathrm{C}\). (b) With \(\mathrm{A}\) as centre and \(\mathrm{AB}\) as radius draw an arc which cuts the circle at the point \(\mathrm{E}\). (c) Construct a circle with radius \(4 \mathrm{~cm}\). (d) Join AB. (e) Draw two radii \(\overline{\mathrm{OA}}\) and \(\overline{\mathrm{OB}}\) such that \(\angle \mathrm{AOB}=72^{\circ}\) (f) Join \(\mathrm{AE}, \mathrm{ED}, \mathrm{DC}\) and \(\mathrm{CB}\). (g) With \(\mathrm{E}\) as centre and \(\mathrm{AB}\) as radius draw an arc which cuts the circle at the point \(\mathrm{D}\). (1) cedgbaf (2) cebdgaf (3) cedbgaf (4) cebgadf

Short answer

Question: Arrange the steps in constructing a regular pentagon with a circle of radius 4 cm in the correct order. Answer: (c), (e), (d), (b), (g), (a), (f)

Step-by-step solution

Correct Order of Steps

Based on the given choices, the correct order of steps is (2), which is: (c) Construct a circle with radius 4 cm. (e) Draw two radii OA and OB such that angle AOB=72°. (d) Join AB. (b) With A as the center and AB as radius draw an arc which cuts the circle at the point E. (g) With E as center and AB as radius draw an arc which cuts the circle at the point D. (a) With D as center and AB as radius draw an arc which cuts the circle at the point C. (f) Join AE, ED, DC, and CB. Let's break down each step:

Construct the Circle

Firstly, draw a circle with the compass set to a radius of 4 cm. Let O be the center of the circle.

Draw Radii OA and OB

Using a protractor, draw two radii OA and OB such that angle AOB is equal to 72°. The angle should be equal to 360°/5, as there are five equal angles in a regular pentagon.

Join AB

Connect points A and B to form the first side of the regular pentagon.

Find Point E

Adjust the compass to match the length of AB by placing its pointed end on A and its pencil end on B. Now, keeping the compass setting, place the pointed end on A and draw an arc that intersects the circle to find the point E.

Find Point D

Keeping the same compass setting with the pointed end on E, draw an arc that intersects the circle to find the point D.

Find Point C

Again, keep the same compass setting and place the pointed end on D to draw an arc that intersects the circle to find point C.

Connect the Points

Join points AE, ED, DC, and CB to form the sides of the regular pentagon inside the circle. Now we have successfully constructed a regular pentagon within a circle of radius 4 cm.

Key concepts

Geometry

Geometry is the branch of mathematics that deals with the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. In the context of constructing shapes, geometry provides the foundational principles that govern how to precisely create various geometrical figures, such as triangles, squares, and in this case, a pentagon.

When constructing a regular pentagon, geometry dictates that all sides and angles must be equal. This requirement comes from the definition of a regular polygon itself. For a pentagon, this means five equal-length sides and five angles, each measuring 72 degrees, which is obtained by dividing the total degrees of a circle, 360 degrees, by the number of sides. Understanding these geometrical principles is essential for accurately completing a construction task.

Circle Constructions

Circle constructions are fundamental in creating geometric shapes within a circular boundary, a common one being the regular pentagon. The entire process of circle constructions for a pentagon involves using the circle's radius as a key measurement, which serves as a guide to space out the pentagon's vertices evenly along the circumference.

The intersection of arcs and the circumference are used strategically to locate the precise points that become the vertices of the pentagon. These points must be equidistant from each other along the edge of the circle, which requires careful use of the compass to maintain consistent measurements.

Geometric Constructions

Geometric constructions are the methods and steps used to create geometric figures without the assistance of measuring devices like rulers or protractors. Instead, these constructions rely on the use of a compass and a straightedge to maintain accuracy.

This process emphasizes accuracy and precision, often starting with basic shapes like circles and straight lines, but gradually combining them to create more complex figures, like our regular pentagon. Mastery of geometric constructions helps students develop a deeper understanding of mathematical relationships and fosters an appreciation for the meticulous nature of geometry.

Compass and Straightedge Constructions

Compass and straightedge constructions form the heart of classical geometrical constructions. By using only a compass and a straightedge, one can create precise angles, straight lines, and perfect polygons. The compass is used to draw arcs and circles, as well as to transfer distances from one part of the construction to another.

The straightedge, which is essentially a ruler without measurement markings, is used to draw straight lines between points. Both tools work in tandem to enforce the constraints of geometric figures, ensuring, for instance, that the sides of a pentagon constructed inside a circle are equal in length. Understanding how to manipulate these tools to construct a variety of shapes is a fundamental skill in geometry.