Introduction to Euclid's Geometry Test

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Introduction to Euclid's Geometry Test - 6

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Question 1
1 / -0

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
1 / -0

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
1 / -0

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.
Question 11
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Which of the following is false?

A
Line segment AB = Line segment BA

B
Line AB = Ray BA

C
Two parallel lines ultimately intersect each other.

D
Two lines subtending an angle of 1 degree intersect each other.

Solution

'Two parallel lines ultimately intersect each other' is an incorrect statement because parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.
Question 12
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On a definite path P to S, Ravi moves from P to R and Shankar moves from R to S. If they both cover the same distance, then

A

B

C

D

Solution

PR = Distance covered by Ravi
RS = Distance covered by Shankar
As they cover the same distance; PR = RS
Question 13
1 / -0

An origami artist made a ball with a very thin paper. How many dimensions will the ball have?

A
It will have no dimensions as the paper is having no thickness.

B
It will have 2 dimensions.

C
It will have only length with no breadth and thickness.

D
It will have 3 dimensions.

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 14
1 / -0

How many planes can pass through a line and a point which is not lying on the line?

A
0

B
1

C
2

D
3

Solution

Let l be a line and A be a point.

Hence, only one plane can pass through a line and a point not lying on the line. Hence, 1 is the correct option.
Question 15
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If a line A is drawn parallel to a line B, which is making two supplement angles on intersection with another two parallel lines C and D, then the sum of the angles on line A will be

A
equal to the sum of angles on line B

B
an obtuse angle

C
equal to 90 degrees

D
more than the sum of interior angles

Solution

Consider C and D to be the parallel lines through which line B is passing, making two supplement angles that add up to 180 degrees.
Now, as given in the question, line A is parallel to line B; so line A will intersect in the same manner and will have same angles. So, angles on line also add up to 180 degrees and will be equal to sum of angles on line B.
Question 16
1 / -0

A Rubik cube has definite surface with length and breadth. How many dimensions does a Rubik cube have?

A
4

B
3

C
2

D
1

Solution

As a Rubik cube has length, breadth and height, it is a 3-dimensional object.
Question 17
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In the given figure, AB = CD. Which of the following relationship is true?

A
AC = BC + CD

B
AB + BC = CD

C
AB – BC = BD

D
AC = CD – BD

Solution

AB = CD (given)

Adding BC to each side,
AB + BC = BC + CD
AC = BC + CD [AB + BC = AC]
Question 18
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If AB = PR and BC = RS, then the equation AB + BC = PR + RS is true by which of the following axioms?

A
If equals are added to equals, the wholes are equal.

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

C
The whole is greater than the part.

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

Solution

According to the question;
AB = PR
Adding BC to both sides,
AB + BC = PR + BC (If equals are added to equals, the wholes are equal.)
But, BC = RS
So, AB + BC = PR + RS
Question 19
1 / -0

Two parallel lines have

A
unique solution

B
infinitely many solutions

C
no solution

D
two solutions

Solution

Parallel lines are two lines that are always the same distance apart and never touch each other. So, there is no solution.
Question 20
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Which of the Euclid postulates defines the relation ∠1 = ∠3 in the given figure?

A
Postulate 1

B
Postulate 2

C
Postulate 3

D
Postulate 4

Solution

According to Euclid's fourth postulate, all right angles are equal to one another.
As it can be seen clearly in the figure that ∠1 = 90° and ∠3 = 90° too; both are right angles and they must be equal according to Euclid's fourth postulate.
Hence, option 4 is the answer.
Question 21
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Determine whether the given statements are true (T) or false (F):

(i) The whole is always less than the part.
(ii) If x = y and if a = b, then x - a = y - b.
(iii) Two distinct intersecting lines can't be parallel to the same line.

A
(i) - F, (ii) - T, (iii) - F

B
(i) - F, (ii) - F, (iii) - T

C
(i) - F, (ii) - T, (iii) - T

D
(i) - F, (ii) - F, (iii) - F

Solution

(i) False
This statement is false and does not follow the Euclid's axiom that the whole is greater than the part
(ii) True
This statement is true and follows the Euclid's axiom: ''If equals are subtracted from equals, the remainders are equal.''
(iii) True
This statement is true and follows the Euclid's axiom: ''Two distinct intersecting lines cannot be parallel to the same line.''
Question 22
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Fill in the blanks:

(i) A line has _______ dimension(s).
(ii) All ________ angles are equal to one another.
(iii) In a test, Awaab's marks are equal to Delilah's marks. After giving grace marks, marks of both become double. Now Awaab's marks are still equal to Delilah's. This is as per the Euclid's __________ axiom.
(iv) According to the Euclid's axiom, things which _______ one another are equal to one another.

A
(i) one (ii) right (iii) second (iv) coincide with

B
(i) two (ii) acute (iii) third (iv) are opposite to

C
(i) one (ii) obtuse (iii) third (iv) are diagonally opposite to

D
(i) two (ii) acute (iii) third (iv) coincide with

Solution

(i) A line has one dimension because only one coordinate is needed to specify a point on it.
(ii) All right angles are equal to one another because all measure 90°.
(iii) According to the Euclid's second axiom , if equals are added to equals, the wholes are equal.
(iv) According to the Euclid's fourth axiom, things which coincide with one another are equal to one another.

Question 23

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Match the following:

Column 1	Column 2
(i) It has no length.	(a) Straight line
(ii) It is a breadth less length.	(b) Surface
(iii) It has only length and breath.	(c) Line
(iv) It lies evenly with the points on itself.	(d) Point

(i) - (a); (ii) - (c); (iii) - (b); (iv) - (d)

(i) - (b); (ii) - (c); (iii) - (a); (iv) - (d)

(i) - (d); (ii) - (c); (iii) - (b); (iv) - (a)

(i) - (c); (ii) - (d); (iii) - (b); (iv) - (a)

Solution

According to definitions given by Euclid,
A point has no length.
A line is a breadth less length.
A surface has only length and breath.
Straight line is a line which lies evenly with the points on itself.

Question 24
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Which statement is correct according to Postulate 5?

A
If a straight line falling on two straight lines makes the interior angles on the same side of it taken together more 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.

B
If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than one right angle, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than one right angle.

C
If a straight line falling on two straight lines makes the interior angles on the opposite side of it taken together with two angles (each 30 degrees), then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

D
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.

Solution

According to Euclid's fifth postulate, '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 25
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Which of the following statements is correct?

(I) If a + b = 5c, b + c = 5d and c = d, then a = d.
(II) If a = b, c = 2d and m = 3n, then a + c - m = b + 2d + 3n.
(III) If a = 2p, b = 2p and c = 2p, then a + b = 2b + c.
(IV) If a + b = 2r and m + n = 2r, then 2a + b = m + 2n.

A
(I)

B
(II)

C
(III)

D
(IV)

Solution

(I) a + b = 5c … (1)
b + c = 5d ... (2)
c = d … (3)
From (1) and (2),
a - c = 5c - 5d
a - d = 5d - 5d [c = d from (3)]
a - d = 0
a = d (True)
Hence, statement I is correct.
(II) If a = b, c = 2d and m = 3n
Adding first two equations,
a + c = b + 2d … (1) [If equals are added to equals, then the wholes are equal.]
Now, subtracting the third equation (m = 3n) from (1),
a + c - m = b + 2d - 3n
Hence, statement II is incorrect.

(III) If a = 2p … (1)
b = 2p … (2)
c = 2p … (3)
Adding (1) and (2),
a + b = 4p … (4)

Adding (2) and (3),
b + c = 4p … (5)
a + b = b + c [Things which are equal to the same thing are equal to one another.]
Hence, statement III is incorrect.

(IV) If a + b = 2r and m + n = 2r
a + b = m + n
[By Euclid's axiom, things which are equal to the same thing are equal to one another.]
Hence, statement IV is incorrect.

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Introduction to Euclid's Geometry Test - 6

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Introduction to Euclid's Geometry Test - 6

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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

Question 11

Line segment AB = Line segment BA

Line AB = Ray BA

Two parallel lines ultimately intersect each other.

Two lines subtending an angle of 1 degree intersect each other.

Solution

Question 12

Solution

Question 13

It will have no dimensions as the paper is having no thickness.

It will have 2 dimensions.

It will have only length with no breadth and thickness.