Chapter 5: Introduction to Euclid’s Geometry

EXERCISE 5.1

1. Which of the following statements are true and which are false ? Given reasons for your answers .

(i) Only one line can pass through a single point .

(ii) There are an infinite number of lines which pass through two distinct points .

(iii) A terminated line can be produced indefinitely on both the sides .

(iv) If two circles are equal , then their radii are equal .

(v) In fig. 5.9 , if AB=PQ and P��=XY , then AB=XY .

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Solution : (i) False . Because an infinite number of lines can pass through a single point.

 (ii) There are an infinite number of lines which pass through two distinct points .

False . Since there are infinitely many points in any given direction from two distinct points, there are an infinite number of lines that can be drawn between those two points.

(iii) A terminated line can be produced indefinitely on both the sides .

True. Because a terminated line can be produced indefinitely .

(iv) If two circles are equal, then their radii are equal.

True. Two circles are equal if they have the same radius. The radius of a circle is the distance from the center of the circle to any point on the circle's circumference. If two circles have the same radius, they will have the same size and shape.

(v) True  .

If two things are equal to the same thing, then they are equal to each other. In this case, AB is equal to PQ, and PQ is equal to XY. Therefore, by transitivity, AB is also equal to XY.

2. Give a definition for each of the following  term . Are there other terms that need to be defined first ? What are they and how might you define them ?

  (i) parallel lines     (ii) perpendicular lines   (iii) line segment  (iv) radius of a circle  (v) square .

Solution:

(i) Parallel lines: Definition: Parallel lines are lines in a two-dimensional plane that never intersect, no matter how far they are extended. They have the same slope and will always maintain a constant distance between each other.

Other terms that need to be defined first:

Lines: A line is a straight one-dimensional figure that extends infinitely in both directions.

(ii) Perpendicular lines: Definition: Perpendicular lines are two lines that intersect at a right angle (90 degrees). The slopes of perpendicular lines are negative reciprocals of each other.

Other terms that need to be defined first:

Lines: As mentioned before.

Intersection: The point where two or more lines meet or cross each other.

(iii) Line segment: Definition: A line segment is a part of a line that is bounded by two distinct endpoints. It is a finite portion of a straight line.

Other terms that need to be defined first:

Lines: As mentioned before.

Endpoints: The points at the beginning and end of a line segment that mark its boundaries.

(iv) Radius of a circle: Definition: The radius of a circle is the distance from the center of the circle to any point on the circumference (outer edge) of the circle.

Other terms that need to be defined first:

Circle: A circle is a two-dimensional geometric shape that consists of all points in a plane that are equidistant from a fixed center point.

(v) Square: Definition: A square is a four-sided polygon with equal sides and right angles at each corner. It is a special type of rectangle with all sides having the same length.

Other terms that need to be defined first:

Polygon: A polygon is a closed two-dimensional shape with straight sides.

Rectangle: A rectangle is a four-sided polygon with opposite sides being equal and all angles being right angles.

Right Angle: An angle that measures exactly 90 degrees.

Side (of a polygon): One of the line segments that form the edges of a polygon.

Note: While some terms were defined within the explanations of others, it's always beneficial to provide clear definitions for the fundamental terms used in mathematics, such as "lines," "intersection," "endpoints," "circle," "polygon," "rectangle," and "right angle."