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Q.4: If a point C lies between two points A and B such that AC = BC, then prove that AC \(=\frac{1}{2} \mathrm{AB}\). Explain by drawing the figure.
Ans : It is given that,
AC = BC
Here, (BC + AC) coincides with AB. It is known that things which coincide with one another are equal to one another. It is also known that things which are equal to the same thing are equal to one another.
Therefore, from equations (1) and (2), we obtain
AC+AC = AB
2AC = AB
Q.5: In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.
Ans :
AC = CB
AC+AC= BC+AC (Equals are added on both sides) …….(1)
Here, (BC + AC) coincides with AB. It is known that things which coincide with one another are equal to one another.
Therefore BC + AC = AB ……(2)
It is also known that things which are equal to the same thing are equal to one another. Therefore, from equations (1) and (2), we obtain
AC + AC = AB
2AC = AB….. (3)
Similarly, by taking D as the mid-point of Ad, it can be proved that
2AD = AB…. (4)
From equation (3) and (4), we obtain
2AC = 2AD (Things which are equal to the same thing are equal to One another.)
AC = AD (Things which are double of the same things are equal to one another.)
This is possible only when point C and D are representing a single point.
Hence, our assumption is wrong and there can be only one mid-point of a given line Segment.
Q.6: In Fig, if AC = BD, then prove that AB = CD.
Ans : From the figure, it can be observed that
AC = AB + BC
BD = BC + CD
It is given that AC = BD
AB + BC = BC + CD(1)
According to Euclid's axiom, when equals are subtracted from equals, the remainders are also equal.
Subtracting BC from equation (1), we obtain
AB + BC - BC = BC + CD - BC
AB = CD
Q.7: Why is Axiom 5, in the list of Euclid's axioms, considered a 'universal truth'? (Note that the question is not about the fifth postulate.)
Ans : Axiom 5 states that the whole is greater than the part. This axiom is known as a universal truth because it holds true in any field, and not just in the field of mathematics. Let us take two cases — one in the field of mathematics, and one other than that.
Case I
Let t represent a whole quantity and only a, b, c are parts of it.
t = a + b + c
Clearly, t will be greater than all its parts a, b, and C.Therefore, it is rightly said that the whole is greater than the part.
Case II
Let us consider the continent Asia. Then, let us consider a country India which belongs to Asia. India is a part Of Asia and it can also be observed that Asia is
greater than India. That is why we can say that the whole is greater than the part. This is true for anything in any part of the world and is thus a universal truth.
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