Introduction to Euclid's Geometry Test

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Introduction to Euclid's Geometry Test - 1

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Question 1
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Which of the following statements is correct regarding the number of dimensions of a sphere?

A
A sphere has 3 dimensions.

B
A sphere can have only 2 dimensions as it is related to a circle.

C
A sphere only has one dimension as it has only one face.

D
A sphere can have both 2 and 3 dimensions as it is related to a circle.

Solution

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in a three-dimensional space.
Question 2
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Which of the following is true about two parallel lines?

A
They cannot have a similar point.

B
They cannot be in the same ratio.

C
They cannot be in the ratio 2 : 3.

D
Both 2 and 3

Solution

Parallel lines are two lines that are always the same distance apart and never touch each other. In order for two lines to be parallel, they must be drawn in the same plane, a perfectly flat surface like a wall or sheet of paper. So, there is no similar point.
Question 3
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If in a line segment PS, a point R bisects the line, then what is the value of PR + RS?

A
2PR

B
2PS

C
SR

D
Half of PS

Solution

R bisects PS (as a bisector divides a line into two equal parts).
So, PR = RS ... (i)
Now, PR + RS
= PR + PR (From (i))
= 2PR
Question 4
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Which of the following has/have two fixed points?

A
Line segment

B
Ray

C
Line

D
Both 1 and 2

Solution

A line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.
Question 5
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Which of the following is true about Euclid's postulate 2?

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

B
All right angles are congruent.

C
A straight line segment cannot be drawn by joining any two points.

D
A line can be drawn by using at least three points.

Solution

Euclid's postulates are as follows:

1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.
4. All right angles are congruent.
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, 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.

So, the second postulate is 'Any straight line segment can be extended indefinitely in a straight line'.
Question 6
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If AB = BC and BC = DF, then which of the following is/are true?

A
AC = DF

B
AB = DF

C
BF = BC

D
Both 2 and 3

Solution

Given: AB = BC and BC = DF
If AB = BC, then AB can be written in the place of BC.
So, AB = DF (As according to Euclid's axiom, "Things which are equal to the same thing are also equal to one another")
Question 7
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On a line AB , C is a point such that : 2AC is equal to AC + CB, then which of the following statements is true?

A
C is the bisector of line AB.

B
C divides AB in the ratio 2 : 3.

C
AC is the double of AB.

D
AC is a quarter of AB.

Solution

AC + CB = AB (By using additive property)
Also, we know that AC + CB = 2AC; which is only possible in the case if AC = CB, which implies that C is the bisector of AB.
Question 8
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Which of the following statements is true according to Euclid's axioms?

(a) If equals are added to equals, the wholes are equal.
(b) If equals are added to equals, the halves are equal.
(c) If equals are added to equals, the parts are equal.
(d) If equals are added to equals, the doubles are equal.

A
(a)

B
(b)

C
(c)

D
(d)

Solution

According to Euclid's second axiom, if equals are added to equals, the wholes are equal.
For instance, if the length of line AB is 2 m and equal to DF and the length of line BC is 2 m and equal to FG; then
AB = BC = 2 m
AB + BC = AC = 4 m
DF + FG = DG = 4m
Thus, if equals are added to equals, the wholes are equal.
Question 9
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Which of the following is not a Euclid's axiom?

A
Things which are equal to the same thing are not necessarily equal to one another.

B
Things which are equal to the same thing are also equal to one another.

C
Things which coincide with one another are equal to one another.

D
If equals be subtracted from equals, the remainders are equal.

Solution

Things which are equal to the same thing are not necessarily equal to one another. This is not a Euclid axiom because the correct axiom is as: Things which are equal to the same thing are also equal to one another.
If AC = BD and BF = BD, then AC = BD = BF
This implies, AC = BF
Question 10
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According to Euclid, which of the following has less than 2 dimensions?

A
Line

B
Square

C
Rectangle

D
Cube

Solution

A line has only 1 dimension, whereas a square and rectangle has 2 dimensions. A cube is a 3-dimensional figure.Thus, only line has less than 2 dimensions.

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Introduction to Euclid's Geometry Test - 1

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Introduction to Euclid's Geometry Test - 1

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Question 1

A sphere has 3 dimensions.

A sphere can have only 2 dimensions as it is related to a circle.

A sphere only has one dimension as it has only one face.

A sphere can have both 2 and 3 dimensions as it is related to a circle.

Solution

Question 2

They cannot have a similar point.

They cannot be in the same ratio.

They cannot be in the ratio 2 : 3.

Both 2 and 3

Solution

Question 3

2PR

2PS

SR

Half of PS

Solution

Question 4

Line segment

Ray

Line

Both 1 and 2

Solution

Question 5

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

All right angles are congruent.

A straight line segment cannot be drawn by joining any two points.

A line can be drawn by using at least three points.

Solution

Question 6

AC = DF

AB = DF

BF = BC

Both 2 and 3

Solution

Question 7

C is the bisector of line AB.

C divides AB in the ratio 2 : 3.

AC is the double of AB.

AC is a quarter of AB.

Solution

Question 8

(a)

(b)

(c)

(d)

Solution

Question 9

Things which are equal to the same thing are not necessarily equal to one another.

Things which are equal to the same thing are also equal to one another.

Things which coincide with one another are equal to one another.

If equals be subtracted from equals, the remainders are equal.

Solution

Question 10

Line

Square

Rectangle

Cube

Solution

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