Rectilinear Figures

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Introduction to Rectilinear Figure

Rectilinear typically indicates a figure representative of making a straight line or along a straight line or in a straight line. In mathematical terms, it is a plane figure bounded by line segments. In other words, a plane figure wholly composed of line segments is known as a rectilinear figure. Now if you are wondering what is a plane figure? Then, know that when we put the tip of a pencil on a sheet of paper and move from one point to the other, without picking up the pencil, then the shapes so formed are what we call as plane curves.


Examples of Rectilinear Figures

Look below to find out the various forms of rectilinear figures. These are as follows:

  1. Quadrilaterals

A quadrilateral can be defined as a 2- dimensional, closed shape geometrical object which consists of four straight sides. Quadrilaterals are also of various types. Most common quadrilaterals include:

  • Square

  • Parallelogram

  • Rectangle

  • Rhombus

  • Trapezium

  • Kite

  1. Polygon

Polygon is a type of a closed rectilinear figure which is also called a simple closed plane curve. The line segments by which a polygon is bounded are called its sides, the points of bisection of consecutive sides known as its vertices, and the angle formed by the meeting of two consecutive sides, is known as its interior angle or simply an angle. Also, know that a polygon is named as per the number of sides it has. For example, A polygon having 3 sides is called a triangle, 4 sides is a Quadrilateral, 5 sides is a Pentagon, 6 sides is a Hexagon, 7 is a Heptagon, 8 is an Octagon and 10 sides is a Decagon.


Types of Polygon

  1. Convex Polygon: If all the (interior) angles of a polygon measure less than 180°, it is known as a convex polygon. In the figure below, it is a convex polygon. It is actually a convex hexagon.

  1. Concave Polygon: If one or more of the (interior) angles of a polygon measures greater than 180° i.e. reflex, it is known as a concave polygon. We also call it as a reentrant polygon.


Key Points to Remember

  • In a convex polygon, the sum of the interior angles of ‘n’ sides is (2n-4) right angles or (2n-4) * 90 degrees.

  • In a regular polygon with n sides (n ≥ 3), then each of its interior angles will be equal to {2n-4}/{n} * 90 degrees.

  • In a regular polygon with n sides, each exterior angle is equal to ({360}/{n})degrees.

  • In a convex polygon, the sum of all exterior angles formed by generating the sides in the same order is equivalent to 4 right angles (or 360 degrees).

Solved Examples

Example: In a pentagon named MNOPQ, MN is parallel to PO and ∠M:∠P:∠Q = 3:4:5 Find ∠Q.

Solution:

Let the measure of the angles of the given pentagon be 3x, 4x and 5x

Therefore,

∠M + ∠N + ∠O + ∠P + ∠Q. = 540 degree

3x + (∠N + ∠O) + 4x + 5x = 540 degree

12x + 180° = 540°

12x = 360°

x = 30°

Hence, the measure of angle will be 5 * 30 = 150°


Example: When you are given the diagonals of a parallelogram that are equal, prove that it is a rectangle.

Solution: Let PQRS be a parallelogram in which PR = QS. We require proving that

∠P = 90°.

In ΔPQR and ΔQPS,

QR = PS (opposite sides of a parallelogram)

PQ = PQ (common)

PR = QS (given, diagonals are equal)

∴ ΔPQR ≅ ΔQPS (as per the SSS rule of congruence)

∴ ∠Q = ∠P (c.p.c.t.)

Since PS || QR and PQ is a transversal,

∠P + ∠Q = 180° (sum of co-int. angles)

⇒ ∠P + ∠P = 180° ⇒ 2∠P = 180° ⇒ ∠P = 90°.

Therefore, PQRS is a rectangle.

FAQ (Frequently Asked Questions)

Q1. What are the Different Types of Plane Curves?

Answer: Plane curves are basically classified into 2 types i.e. open curve and closed curve. The (plane) curves that consist of different starting and endpoints are known as open curves while the curves that consist of the same starting and endpoints are called closed curves. However, there is one more type of plane curve which is called a simple curve. It simply makes for a curve that does not cross itself at any point is called a simple curve.

Q2. What is an Example of a Simple Closed Plane Curve?

Answer: A simple closed plane curve is that which is entirely made up of line segments. A polygon is a simple closed plane curve.

Q3. What is Diagonal of a Polygon?

Answer: A Diagonal of a polygon is basically a line segment connecting any two nonconsecutive vertices of a polygon.

Q4. What are the Properties of a Diagonal of a Polygon?

Answer: With different types of polygons, there are several properties as follows:

  • In a parallelogram, both the pairs of opposite sides and opposite angles are equal.

  • In a quadrilateral, if one pair of opposite sides is equal and parallel, the figure so formed is a parallelogram.

  • The diagonals of a parallelogram intersect each other and each diagonal of a parallelogram intersect the parallelogram

  • A special parallelogram called Rhombus is a polygon whose diagonals meet at right angles.

  • In a square and rectangle, diagonals are equal.