# The Elements of Geometry

Chapters

- ap-scert-9th-class-Maths-Lesson-1-Real-Numbers
- ap-scert-9th-class-Maths-Lesson-2-Polynomials-and-Factorisation
- ap-scert-9th-class-Maths-Lesson-3-The-Elements-of-Geometry
- ap-scert-9th-class-Maths-Lesson-4-Lines-and-Angles
- ap-scert-9th-class-Maths-Lesson-5-Co-Ordinate-Geometry
- ap-scert-9th-class-Maths-Lesson-6-Linear-Equation-in-Two-Variables
- ap-scert-9th-class-Maths-Lesson-7-Triangles
- ap-scert-9th-class-Maths-Lesson-8-Quadrilaterals
- ap-scert-9th-class-Maths-Lesson-9-Statistics
- ap-scert-9th-class-Maths-Lesson-10-Surface-Areas-and-Volumes
- ap-scert-9th-class-Maths-Lesson-11-Areas
- ap-scert-9th-class-Maths-Lesson-12-Circles
- ap-scert-9th-class-Maths-Lesson-13-Geometrical-Constructions
- ap-scert-9th-class-Maths-Lesson-14-Probability
- ap-scert-9th-class-Maths-Lesson-15-Proofs-in-Mathematics

- ap-scert-9th-class-Maths-Lesson-1-Real-Numbers
- ap-scert-9th-class-Maths-Lesson-2-Polynomials-and-Factorisation
- ap-scert-9th-class-Maths-Lesson-3-The-Elements-of-Geometry
- ap-scert-9th-class-Maths-Lesson-4-Lines-and-Angles
- ap-scert-9th-class-Maths-Lesson-5-Co-Ordinate-Geometry
- ap-scert-9th-class-Maths-Lesson-6-Linear-Equation-in-Two-Variables
- ap-scert-9th-class-Maths-Lesson-7-Triangles
- ap-scert-9th-class-Maths-Lesson-8-Quadrilaterals
- ap-scert-9th-class-Maths-Lesson-9-Statistics
- ap-scert-9th-class-Maths-Lesson-10-Surface-Areas-and-Volumes
- ap-scert-9th-class-Maths-Lesson-11-Areas
- ap-scert-9th-class-Maths-Lesson-12-Circles
- ap-scert-9th-class-Maths-Lesson-13-Geometrical-Constructions
- ap-scert-9th-class-Maths-Lesson-14-Probability
- ap-scert-9th-class-Maths-Lesson-15-Proofs-in-Mathematics

- Solutions
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Solutions

Answer the following:

i) How many dimensions a solid has ?

Solution:

A solid has three dimensions namely length, breadth and height or depth.

ii) How many books are there in Euclid’s Elements ?

Solution:

There are 13 volumes in Euclid’s elements.

iii) Write the number of faces of a cube and cuboid.

Solution:

Cube : 6 faces

Cuboid : 6 faces

iv) What is the sum of interior angles of a triangle ?

Solution:

The sum of interior angles of a triangle is 180°.

v) Write three undefined terms of geometry.

Solution:

Point, line and plane are three undefined terms in geometry.

State whether the following statements are true or false. Also give reasons for your answers.

- a) Only one line can pass through a given point
- b) All right angles are equal
- c) Circles with same radii are equal
- d) A finite line can be extended on its both sides endlessly to get a straight line

e) From figure AB > AC

Solution:

a) Only one line can pass through a given point - False.

Reason : (Since, infinitely many lines can pass through a given point)

b) All right angles are equal - True.

c) Circles with same radii are equal - True.

d) A finite line can be extended on its both sides endlessly to get a straight line - True.

e) From figure AB > AC - True.

In the figure given below, show that the length AH > AB + BC + CD.

Solution:
Given a line AH↔

To prove AH > AB + BC + CD

From the figure AB + BC + CD = AD

AD is a part of whole AH.

From Euclid’s axiom whole is greater than part.

∴ AH > AD

⇒ AH > AB + BC + CD

If a point Q lies between two points P and R such PQ = QR, prove that PQ = 1/2 PR.

Let PR be a given line.

Given that PQ = QR

i. e., Q is a point on PR.

⇒ PQ + QR = PR

⇒ PQ + PQ = PR [∵ PQ = QR]

⇒ 2PQ = PR

⇒ PQ = 1/2 PR

Hence proved.

Draw an equilateral triangle whose sides are 5.2 cm

Soluton:

- Step 1 : Draw a line segment AB of length 5.2 cm. *
- Step 2 : Draw an arc of radius 5.2 cm with centre A.
- Step 3 : Draw an arc of radius 5.2 cm with centre B.
- Step 4 : Two arcs intersect at C; join C to A and B.

Δ ABC is the required triangle.

What is a conjecture? Give an example for it.

Solution:

Mathematical statements which are neither proved nor disproved are called conjectures. Mathematical discoveries often start out as conjectures. This may be an educated guess based on observations.

Eg : Every even number greater than 4 can be written as sum of two primes. This example is called Gold Bach Conjecture

Mark two points P and Q. Draw a line through P and Q. Now how many lines are parallel to PQ, can you draw ?

Solution:

Infinitely many lines parallel to PQ can be drawn.

In the figure given below, a line n falls on lines / and m such that the sum of the interior angles 1 and 2 is less than 180°, then what can you say about lines l and m ?

Solution:

Given : l, m and n are lines, n is a transversal.

∠1 < 90°

∠2 < 90°

If the lines l and m are produced on the side where angles 1 and 2 are formed, they intersect at one point.

In the figure given below, if ∠1 = ∠3, ∠2 = ∠4 and ∠3 = ∠4 write the rela-tion between ∠1 and ∠2 using Euclid’s postulate.

Solution :

Given : ∠1 = ∠3

∠3 = ∠4

∠2 = ∠4

∴∠1 = ∠2

∵Both ∠1 and ∠2 are equal to ∠4. (By Euclid’s axiom things which are equal to same things are equal to one another).

In the figure given below, we have BX = 1/2 AB, BY= 1/2 BC and AB = BC. Show that BX = BY.

Solution:

Given : BX = 12 AB

BY = 12BC

AB = BC

To prove : BX = BY

Proof: Given AB = BC [ ∵ By Euclid’s axiom things which are halves of the same things are equal to one another]

12 AB = 12 BC

BX = BY

Hence proved.

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