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Euclid, an Egyptian mathematics teacher in 300 BCE, collected, rearranged and redefined the very geometrical concepts and theories and formulated them in his own way which greatly influenced the way geometry was conceived for the ages to come.
NCERT Class-9 Chapter-5 Introduction to Euclid’s Geometry discusses Euclid’s Definitions, Axioms, Postulates related to geometry.
The Geometry section of the Class-9 NCERT question paper comprises 8 questions carrying a total of 22 marks.
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The following article presents a few Multiple Choice Questions related to the Class-9 Chapter-5 Introduction to Euclid’s Geometry
Question 1: Which of the following statements are true?
- Only one line can pass through a single point.
- There is an infinite number of lines that pass through two distinct points.
- A terminated line can be produced indefinitely on both sides.
- If two circles are equal, then their radii are unequal.
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Answer: c
Explanation
The first statement is false because we can draw an infinite number of lines through a given point. The second statement is also false because, according to Axiom 5.1, Given two distinct points, there is a unique line that passes through them. The third statement is true because, according to Postulate 2, a terminated line can be produced indefinitely. The fourth statement is false because, according to Postulate 3, a circle can be drawn with any center and radius. We know that circles are equal. That means the circles are congruent, which means that circumferences are equal and so the radii of the two circles are also equal.
Question 2: A solid has __________dimensions.
- One
- Two
- Three
- Zero
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Answer: c
Explanation
Consider three steps from solid to point (solid-surface-line-point). At each step we lose an extension, also called a dimension. Dimension is the measure of something in a certain direction and the length, breadth, and height of the solid. So, a solid has three dimensions, a surface has two, a line has one and a point has none.
Question 3: A point has _______ dimension.
- One
- Two
- Three
- Zero
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Answer: d
Explanation
We know that the dimension of a point is zero because we cannot determine the size of a point. In other words, we can say that a point is an exact position. It has no shape but only one position. A point is a place, not a thing. So there is no dimension to it.
Question 4: The shape of the base of the Pyramid is:
- Triangle
- Square
- Rectangle
- Any Polygon
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Answer: d
Explanation
A pyramid is a structure whose outer surfaces are triangular and converge at a single point at the top. The base of a pyramid can be in the shape of a triangle, it can be square or some other polygon. So, option (d) is the correct answer to this question.
Question 5: The boundaries of solid are called:
- Surfaces
- Curves
- Lines
- Points
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Answer: a
Explanation
Solid can be a shape, size, position, and moved from one place to another. Its boundary is called the surface. They separate one part of space from another and are said to have no thickness. The boundaries of the surfaces are curved or straight lines. These lines end at the point.
Question 6: A surface of a shape has:
- Length, breadth, and thickness
- Length and breadth only
- Length and thickness only
- Breadth and thickness only
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Answer: b
Explanation
Dimension is the measure of something in a certain direction and the length, breadth, and height of the solid. So, a surface is a two-dimension plane without any volume according to Euclid. Hence, it has only length and breadth but no thickness. So, option (b) is the correct answer.
Question 7: The edges of the surface are:
- Points
- Curves
- Lines
- None of the above
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Answer: c
Explanation
Solid can be a shape, size, position, and moved from one place to another. Its boundary is called the surface. A surface is that which has length and breadth only. The edges of a surface are lines. A point is that which has no part. A line is a breadthless length. The ends of a line are points.
Question 8: Which of these statements does not satisfy Euclid’s axiom?
- Things that are equal to the same thing are equal
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- The whole is lesser than the part.
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Answer: d
Explanation
The first three statements are satisfied Euclid’s axiom. Only the fourth statement doesn’t satisfy Euclid’s axiom. According to Euclid’s axiom “The whole is greater than the part”. For example, if a quantity N is a part of another quantity M, then M can be written as the sum of N and some third quantity P. Symbolically, M > N means that there is some P such that M = N + P.
Question 9: The line drawn from the center of the circle to any point on its circumference is called:
- Radius
- Diameter
- Sector
- Arc
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Answer: a
Explanation
Radius is a line segment joining the center of the circle and at any point on the boundary of the circle. Dimension is the measure of something in a certain direction and the length, breadth, and height of the solid. A piece of a circle between two points is called Arc. The region between an arc and the two radii, joining the center to the endpoints of the arc is called a sector. So, the radius is the correct answer.
Question 10: There are ________ number of Euclid’s Postulates
- Three
- Four
- Five
- Six
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Answer: c
Explanation
There are five Euclid’s Postulates. They are:
- Postulate 1: A straight line may be drawn from any point to any other point.
- Postulate 2: A terminated line can be produced indefinitely.
- Postulate 3: A circle can be drawn with any center and radius.
- Postulate 4: All right angles are equal.
- Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
Question 11: Euclid’s fifth postulate is:
- The whole is greater than the part
- All right angles are equal
- If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines if produced indefinitely, meet on that side on which the sum of angles is less than two right angles
- A circle may be described with any center and any radius
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Answer: c
Explanation
Euclid's fifth hypothesis is that if a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
Question 12: The three steps from solids to points are:
- Lines – points – surfaces – solids
- Lines – surfaces – points – solids
- Solids – lines – surfaces – points
- Solids – surfaces – line – points
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Answer: d
Explanation
A solid has shape, size, position, and can be moved from one place to another. Its boundaries are called surfaces. They separate one part of the space from another and are said to have no thickness. The boundaries of the surfaces are curved or straight lines. These lines end in points.
Question 13: Axioms are assumed:
- Theorems
- Definitions
- Universal truths specific to geometry
- Universal truths in all branches of mathematics
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Answer: d
Explanation
Starting with his definitions, Euclid assumed certain properties, which were not to be proved. These assumptions are actually “obvious universal truths”. He divided them into two types: axioms and postulates. He used the term ‘postulate’ for the assumptions that were specific to geometry. Common notions (often called axioms), on the other hand, were assumptions used throughout mathematics and not specifically linked to geometry. So, axioms are assumed universal truths in all branches of mathematics.
Question 14: It is known that if x + y = 10 then x + y + z = 10 + z. Euclid’s axiom that illustrates this statement is
- First Axiom
- Second Axiom
- Third Axiom
- Fourth Axiom
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Answer: b
Explanation
According to the Second Axiom, If equals are added to equals, the wholes are equal. Hence, if z has been added to both the sides of equation x + y = 10, then it becomes x + y + z = 10 + z.
Question 15: Euclid stated that all right angles are equal to each other in the form of
- Definition
- Proof
- Postulate
- Axiom
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Answer: c
Explanation
The correct answer is a postulate because Euclid stated that all right angles are equal to each other in the form of the fourth postulate. According to his statement, one line perpendicular to another can help make the right angle and this makes one angle equal to another.
Question 16: ‘Lines are parallel if they do not intersect’ is stated in the form of
- Definition
- Proof
- Postulate
- Axiom
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Answer: a
Explanation
‘Lines are parallel if they do not intersect’ is stated in the form of definition. The definition is a statement that gives the exact meaning of the word.
Question 17: The side faces of a pyramid are:
- Square
- Triangles
- Polygons
- Trapezium
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Answer: b
Explanation
The side faces of a pyramid are triangles.
Question 18: Select the wrong statement:
- Only one line can pass through a single point.
- Only one line can pass through two distinct points.
- A terminated line can be produced indefinitely on both sides.
- If two circles are equal, then their radii are unequal.
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Answer: a
Explanation
Only one line can pass through a single point.
Question 19: The total number of propositions in Euclid’s famous treatise “The Elements” is:
- 13
- 55
- 460
- 465
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Answer: d
Explanation
Euclid deduced 465 propositions in a logical chain using his axioms, postulates, definitions, and theorems proved earlier in the chain.
Question 20: The things which are double of the same thing are
- equal
- unequal
- double of the same thing
- halves of the same thing
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Answer: a
Explanation
According to Euclid’s axiom, the things which are double the same thing are equal to each other.
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