If $A$,$B$ and $C$ are three points on a line,and $B$ lies between $A$ and $C$ (see figure),then prove that $AB + BC = AC$.
(N/A) In the figure given above,the line segment $AC$ coincides with the sum of the line segments $AB$ and $BC$.
According to Euclid's Axiom $(4)$,things which coincide with one another are equal to one another.
Since the segment $AB + BC$ coincides with the segment $AC$,it follows that:
$AB + BC = AC$
Note that in this proof,it is assumed that there is a unique line passing through two distinct points.
According to Euclid's Axiom $(4)$,things which coincide with one another are equal to one another.
Since the segment $AB + BC$ coincides with the segment $AC$,it follows that:
$AB + BC = AC$
Note that in this proof,it is assumed that there is a unique line passing through two distinct points.
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Similar Questions
Which of the following statements are true and which are false? Give 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 the figure,if $AB = PQ$ and $PQ = XY,$ then $AB = XY$.
$(i)$ 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 the figure,if $AB = PQ$ and $PQ = XY,$ then $AB = XY$.
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View SolutionProve that an equilateral triangle can be constructed on any given line segment.
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View SolutionIn the figure,if $AC = BD$,then prove that $AB = CD$.
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View SolutionIf a point $C$ lies between two points $A$ and $B$ such that $AC = BC$,then prove that $AC = \frac{1}{2} AB$. Point $C$ is called a mid-point of line segment $AB$. Prove that every line segment has one and only one mid-point.
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