Page 7.12
(i) These exists a point through which no line can pass.
Sol : False : Infinite number of lines cam puss through a given point.
(ii) A terminated line can be produced on either sides indefinitely.
Sol : True : By Euclid's second postulate.
(iii) With a given point A as centre, one and only one circle exists.
Sol : False : With the given point as centre and different radii , infinite number of circles can be drawn
(iv) There exist two numbers x and y such that x ≠ y but 2x = 2y.
Sol : False : Since 2x=2y ⇒ x=y [Divide both sides by 2]
(v) Two different lines, can have at most one point common.
Sol : True : By theorem 7.1
Sol :
In Geometry, "Axiom" and "Postulate" are essentially interchangeable. In antiquity, they referred to propositions that were "obviously true" and only had to be stated, and not proven. In modern mathematics there is no longer an assumption that axioms are "obviously true". Axioms are merely 'background' assumptions we make. The best analogy I know is that axioms are the "rules of the game".
Sol:
A theorem is a logical consequence of the axioms. In Geometry, the "propositions" are all theorems: they are derived using the axioms and the valid rules. A "Corollary" is a theorem that is usually considered an "easy consequence" of another theorem. What is or is not a corollary is entirely subjective. Sometimes what an author thinks is a 'corollary' is deemed more important than the corresponding theorem. (The same goes for "Lemma"s, which are theorems that are considered auxiliary to proving some other, more important in the view of the author, theorem).
Sol : Point , Line and Plane
Sol :
In Euclid's Geometry, the main axioms/postulates are:
Sol :
Sol :
Through any point in the plane, there is at most one straight line parallel to a given straight line. This axiom is equivalent to the parallel postulate.
(i) Parallel lines
Sol :
Two lines l and m are called parallel if they have no common point . They are also equidistant from each other
UNDEFINED TERMS:
(a) Lines : These are taken as undefined.
(b) Point : This is taken as undefined.
(c) Common : Situated on both
(ii) Acute angle
Sol :
An angle which measures less than a right angle.
UNDEFINED TERMS:
(a) Measure
(b) Less than
(c) Right angle
(iii) Rectangle
Sol :
Figure formed by four lines whose all the four angles are right angle.
UNDEFINED TERMS:
(a) Figure formed by four lines
(b) Angle
(c) Equal
(iv) Parallelogram
Sol :
A quadrilateral (figure formed by four lines) in which two line segments are parallel and the other two line segments are also parallel.
UNDEFINED TERMS:
(a) Quadrilateral
(b) Line : This is taken as undifined
Type 2
Sol:
Sol :
⇒Given PR=QS also PQ=QR
⇒PR=QS
[can be written as]
⇒PQ+QR=QR+RS
On subtracting QR from both sides
[If equals are subtracted from from equals , the remainders are equal]
⇒PQ=RS ..(i)
⇒also it is given that PQ=QR..(ii)
[Things which are equal to the same thing are equal to one another]
⇒From (i) and (ii)
⇒PQ=QR=RS..(iii)
Also ,
⇒PS=PQ+QR+RS
⇒PS=PQ+PQ+PQ [from (iii)]
⇒PS=3PQ
⇒$\dfrac{PS}{3}={PQ}$
Hence Proved
Exercise 7.1
Type 1Question 1
Which of the following statements are true and which are false ? Give reasons for your answer :(i) These exists a point through which no line can pass.
Sol : False : Infinite number of lines cam puss through a given point.
(ii) A terminated line can be produced on either sides indefinitely.
Sol : True : By Euclid's second postulate.
(iii) With a given point A as centre, one and only one circle exists.
Sol : False : With the given point as centre and different radii , infinite number of circles can be drawn
(iv) There exist two numbers x and y such that x ≠ y but 2x = 2y.
Sol : False : Since 2x=2y ⇒ x=y [Divide both sides by 2]
(v) Two different lines, can have at most one point common.
Sol : True : By theorem 7.1
Question 2
What do you understand by Axioms and Postulates ?Sol :
In Geometry, "Axiom" and "Postulate" are essentially interchangeable. In antiquity, they referred to propositions that were "obviously true" and only had to be stated, and not proven. In modern mathematics there is no longer an assumption that axioms are "obviously true". Axioms are merely 'background' assumptions we make. The best analogy I know is that axioms are the "rules of the game".
Question 3
What is a theorem ?Sol:
A theorem is a logical consequence of the axioms. In Geometry, the "propositions" are all theorems: they are derived using the axioms and the valid rules. A "Corollary" is a theorem that is usually considered an "easy consequence" of another theorem. What is or is not a corollary is entirely subjective. Sometimes what an author thinks is a 'corollary' is deemed more important than the corresponding theorem. (The same goes for "Lemma"s, which are theorems that are considered auxiliary to proving some other, more important in the view of the author, theorem).
Question 4
Write some terms which are not defined in geometry.Sol : Point , Line and Plane
Question 5
Write Euclid's five postulates.Sol :
In Euclid's Geometry, the main axioms/postulates are:
- Given any two distinct points, there is a line that contains them.
- Any line segment can be extended to an infinite line.
- Given a point and a radius, there is a circle with center in that point and that radius.
- All right angles are equal to one another.
- If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (The parallel postulate).
Question 6
Write Euclid's three axioms.Sol :
- First Axiom: Things which are equal to the same thing are also equal to one another.
- Second Axiom: If equals are added to equals, the whole are equal.
- Third Axiom: If equals be subtracted from equals, the remainders are equal.
Question 7
Write Playfair's axiom.Sol :
Through any point in the plane, there is at most one straight line parallel to a given straight line. This axiom is equivalent to the parallel postulate.
Question 8
Give definition of each of the following terms. Are there some terms which need to be defined ? What are those and how do you define them ?(i) Parallel lines
Sol :
Two lines l and m are called parallel if they have no common point . They are also equidistant from each other
UNDEFINED TERMS:
(a) Lines : These are taken as undefined.
(b) Point : This is taken as undefined.
(c) Common : Situated on both
(ii) Acute angle
Sol :
An angle which measures less than a right angle.
UNDEFINED TERMS:
(a) Measure
(b) Less than
(c) Right angle
(iii) Rectangle
Sol :
Figure formed by four lines whose all the four angles are right angle.
UNDEFINED TERMS:
(a) Figure formed by four lines
(b) Angle
(c) Equal
(iv) Parallelogram
Sol :
A quadrilateral (figure formed by four lines) in which two line segments are parallel and the other two line segments are also parallel.
UNDEFINED TERMS:
(a) Quadrilateral
(b) Line : This is taken as undifined
Type 2
Question 9
In the figure given below, if AB=BC=CD , prove that $\text{AB}=\dfrac{1}{3}\text{AD}$Sol:
⇒AD = AB+BC+CD
[ Given AB=BC=CD ]
⇒AD = AB+AB+AB
⇒AD = 3AB
⇒$\dfrac{\text{AD}}{3}=AB$
Question 10
In the figure given below, if PR=QS and PQ=QR , prove that $\text{PQ}=\dfrac{1}{3}\text{PS}$Sol :
⇒Given PR=QS also PQ=QR
⇒PR=QS
[can be written as]
⇒PQ+QR=QR+RS
On subtracting QR from both sides
[If equals are subtracted from from equals , the remainders are equal]
⇒PQ=RS ..(i)
⇒also it is given that PQ=QR..(ii)
[Things which are equal to the same thing are equal to one another]
⇒From (i) and (ii)
⇒PQ=QR=RS..(iii)
Also ,
⇒PS=PQ+QR+RS
⇒PS=PQ+PQ+PQ [from (iii)]
⇒PS=3PQ
⇒$\dfrac{PS}{3}={PQ}$
Hence Proved
No comments:
Post a Comment