NCERT Class 9 Solutions: Introduction To Euclid's Geometry (Chapter 5) Exercise 5.1 – Part 1

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Euclid's Geometry of circles, lines, triangles

Euclid's Geometry

Euclid's Geometry of circles, lines, triangles

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions. Here are Euclid’s postulates:

  1. A straight line segment can be drawn joining any two points.

    A line segment AB

    A Line Segment AB

    A line segment AB

  2. Any straight line segment can be extended indefinitely in a straight line.

    A line AB continues infinitely in both directions

    A Line

    A line AB continues infinitely in both directions

  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

    Circle with center O and radius r

    Circle With Center O and Radius R

    Circle with center O and radius r

  4. All right angles are congruent.

    Understanding right angle as a quater of a circle

    Right Angle

    Understanding right angle as a quater of a circle

  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles ( 180° ), then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. In the below figure α+β<180° , thus the lines intersect on that side.

    Lines intersect because angle alpha + angle beta is less than 180 degree

    Parallel Postulate

    Lines intersect because angle alpha + angle beta is less than 180 degree

Q-1 Which of the following statement are true and which are false? Give reasons for your answer.

  1. Only one line can pass through a single point.

  2. There are infinite numbers of lines which pass through two distinct points.

  3. A terminated line can be produced on both the sides.

  4. If two circles are equal, then their radii are equal.

  5. In the following figure, if AB=PQ and PQ=XY , than AB=XY

Give the point AB,PQ,XY & AB=PQ AND PQ=XY

Give the Point AB,PQ,XY

Give the point AB,PQ,XY & AB=PQ AND PQ=XY

Solution (1.)

  • False. as we know that there are various points in a plane. Such that A, B, C, D and E. Now by first postulate we know that a line may be drawn from any of these points to all the other points. Giving many lines. In general since there are infinite points in a plane, an infinite number of lines can be drawn from any of these points to other points.

Give Point of A,B,C,D,E it prove that many lines can pass throgh point A.similarly we conclude that infinite lines can pass through a single point

Give Point of a,B,C,D,E

Give Point of A,B,C,D,E it prove that many lines can pass throgh point A.similarly we conclude that infinite lines can pass through a single point

Solution (2.)

Give two point A and B on plane papere observe that there is just only one line passes through both A and B

Give Two Point a and B

Give two point A and B on plane papere observe that there is just only one line passes through both A and B

  • Let us mark two points A and B on the plane of paper. Now we fold the paper so that a crease passes through A. Since we know that an unlimited number of lines can pass through a point. So an unlimited number of lines can pass through A.

  • Again we fold the paper so that a line passes through B. Clearly infinite number of lines can pass through B. Now we fold the paper in such a way that a line passes through both A and B.

  • We observe that there is just only one line passes through both A and B.

  • This is what was told by Euclid’s 1st postulate.

Solution (3.)

  • True

  • In geometry, by a line, we mean the line in its totality and not a portion of it. A physical example of a perfect line is not possible. Since a line extends indefinitely in both the directions.

    Describing a line as an infinite extension

    Understanding the Concept of a Line

    Describing a line as an infinite extension

  • It cannot be drawn or shown whole on paper. In practice, only a portion of a line is drawn and arrowheads are marked at its two ends indicating that it extends indefinitely in both directions

  • This was the second postulate. Note that a line terminated on both sides is known as line segment. A line terminated only on one side is known as a ray.

Solution (4.)

  • True

  • If the circles are of same radii, then we can fully superimpose then by bringing their centers one on top of another. Therefore, their radii will be equal.

    Two congruent circles with centers M and K

    Two Congruent Circles

    Two congruent circles with centers M and K

Solution (5.)

  • True, because things which are equal to the same thing, are equal to one another. This was giving in the Euclid’s axioms:

  1. Things which equal the same thing also equal one another.

  2. If equals are added to equals, then the wholes are equal.

  3. If equals are subtracted from equals, then the remainders are equal.

  4. Things which coincide with one another equal one another.

  5. The whole is greater than the part.

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