Chapter 3: The Elements of Geometry (Exercise 3.1)

Chapter 3: The Elements of Geometry (Exercise 3.1)

Question 1:
Answer the following:
1. How many dimensions does a solid have?
2. How many books are there in Euclid's Elements?
3. Write the number of faces of a cube and a cuboid.
4. What is the sum of interior angles of a triangle?
5. Write three undefined terms of geometry.

Solution:
1. A solid has 3 dimensions (Length, Breadth, and Height).
2. Euclid's Elements consists of 13 books.
3. Both a cube and a cuboid have 6 faces.
4. The sum of the interior angles of a triangle is 180°.
5. The three undefined terms in geometry are Point, Line, and Plane.

Question 2:
State whether the following statements are True or False. Give reasons for your answer:
1. Only one line can pass through a given point.
2. All right angles are equal to one another.
3. Circles with the same radii are equal.
4. A finite line can be extended on both sides endlessly to get a straight line.

Solution:
1. False: An infinite number of straight lines can pass through a single point.
2. True: By Euclid's 4^th Postulate, all right angles are equal to 90° and thus equal to one another.
3. True: If two circles have the same radius, superimposing one over the other will make their centers and boundaries coincide completely.
4. True: By Euclid's 2^nd Postulate, a terminated/finite line segment can be produced indefinitely.

Question 3:
In a line segment AH containing points B, C, D in order, show that AH > AB + BC + CD.

Solution:
* By Euclid's Axiom 5, "The whole is greater than the part."
* Here, AH is the entire line segment (the whole), whereas AB, BC, and CD are parts of line segment AH.
* Since AB + BC + CD = AD, and AD is a part of AH, we get:
AH > AB + BC + CD

Question 4:
If a point Q lies between two points P and R such that PQ = QR, prove that PQ = ½ PR.

Solution:
Given that point Q lies between P and R and PQ = QR.
Line segment PR = PQ + QR

Since QR = PQ, substitute QR with PQ:
PR = PQ + PQ
PR = 2PQ
PQ = ½ PR
(Hence proved.)

Question 5:
Draw an equilateral triangle whose sides are 5.2 cm each.

Solution (Steps of Construction):
1. Draw a line segment AB = 5.2 cm using a ruler.
2. With center A and radius 5.2 cm (Euclid's 3^rd Postulate), draw an arc.
3. With center B and the same radius 5.2 cm, draw another arc intersecting the previous arc at point C.
4. Join A to C and B to C.
5. ΔABC is the required equilateral triangle with side lengths 5.2 cm.

Question 6:
What is a conjecture? Give an example.

Solution:
* Definition: A conjecture is a statement or mathematical proposition that is believed to be true based on observations or empirical evidence, but has not yet been rigorously proven or disproven.
* Example: Goldbach's Conjecture states that "Every even integer greater than 2 can be expressed as the sum of two prime numbers."

Question 7:
Mark two points P and Q. Draw a line passing through P and stream Q. How many lines parallel to line PQ can you draw?

Solution:
* Mark points P and Q and draw a unique line passing through both points.
* An infinite number of lines can be drawn parallel to the line PQ on a two-dimensional plane.

Question 8:
A transversal line n falls on lines l and m such that the sum of interior angles ∠1 + ∠2 < 180° on one side. What can you say about lines l and m?

Solution:
* According to Euclid's 5^th Postulate, if a straight line falling on two straight lines makes interior angles on the same side whose sum is less than two right angles (180°), then the two lines, if extended indefinitely, will eventually meet/intersect on that same side.
* Therefore, lines l and m will intersect on the side where ∠1 + ∠2 < 180°.

Question 9:
If ∠1 = ∠3, ∠2 = ∠4, and ∠3 = ∠4, write the relation between ∠1 and ∠2 using Euclid's Axiom.

Solution:
* Given: ∠1 = ∠3, ∠2 = ∠4, and ∠3 = ∠4.
* By Euclid's Axiom 1: "Things which are equal to the same thing are equal to one another."
* Since ∠1 = ∠3 and ∠3 = ∠4, we get ∠1 = ∠4.
* Since ∠2 = ∠4 and ∠1 = ∠4, we get:
∠1 = ∠2

Question 10:
In a figure, BX = ½ AB, BY = ½ BC, and AB = BC. Show that BX = BY.

Solution:
* Given: AB = BC
* Dividing both sides by 2 (Euclid's Axiom 7: "Things which are halves of the same things are equal to one another"):
½ AB = ½ BC
* Substituting BX = ½ AB and BY = ½ BC:
BX = BY
(Hence proved.)