AP 9th Class Maths Chapter 5 Introduction to Euclid’s Geometry Important Questions

These AP 9th Class Maths Important Questions 5th Lesson Introduction to Euclid’s Geometry will help students prepare well for the exams.

AP Board Class 9 Maths 5th Lesson Introduction to Euclid’s Geometry Important Questions

9th Class Maths Introduction to Euclid’s Geometry 2 Marks Important Questions

Question 1.
In the figure given below, show that the length AH > AB + BC + CD.

Solution:
Given a line \(\overleftrightarrow{\mathrm{AH}}\)
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

Question 2.
Mark two points P and Q. Draw a line through P and Q. Now how many lines which are parallel to PQ, can you draw?
Solution:

Infinitely many lines parallel to PQ can be drawn.

Question 3.
Does Euclid’s fifth postulate imply the existence of parallel lines ? Explain.
Answer:
If a straight line l falls on two straight lines m and n such that sum of the interior angles on one side of l is two right angles, then by Euclid’s fifth postulate the lines m and n will not meet on the side of l. Next, we know that the sum of the interior angles on the other side of line l will also be two right angles.

Therefore, they will not meet on the other side also. So, the lines m and n never meet and are therefore parallel.

Question 4.
Solve the equation x + 25 = 40 and state which axiom you use here. Also give two more axioms other than the axioms used in the above situation.
Solution:
x + 25 = 40
⇒ x + 25 – 25 = 40 – 25
(If equals are subtracted from equals, the remainders are equal.)
⇒ x = 15
Two more axioms:
i) Things which coincide with one another are equal to one another.
ii) Things which are halves of the same things are equal to one another.

Question 5.
Two salesmen make equal sales during the month of June. In July, each salesman doubles his sale of the month of June. Compare their sales in July. State which axiom, you use here? Also give two more axioms other than the axiom used in the above situation.
Solution:
Let the sales of two salesmen in the month of June be a and b.
Then, according to the question,
a = b
2a = 2b
Things which are double of the same things are equal to one another [Euclid’s Axiom (vi)] .
⇒ Their sales in July are equal.
Two more axioms:
i) The whole is greater than the part.
ii) Things which are equal to the same thing are equal to one another.

Question 6.
Write any three Euclid’s postulates.
(OR)
Write any two Euclid’s postulates.
Answer:
i) All right angles are equal to one another.
ii) A straight line may be drawn from any one point to any other point.
iii) A circle can be drawn with any centre and any radius.

Question 7.
Sunil and Shyam have the same weight. If they each gain weight by 5 kg. How ‘ will their new weights be compared using the axioms ? Write the Euclid’s axiom the be$t supports your answer. Also give two more axioms other than the axiom used in the above situation.
Answer:
Their weights are equal.
If equals are added to equals, the wholes are equal [Euclid’s Axiom (ii)]
Two more axioms:
i) The whole is greater than the part.
ii) Things which are halves of the same things are equal to one another.

Question 8.
Write suitable Euclid’s axiom for the statement “If x = y then x + 2 = y + 2”.
Solution:
If x = y then x + 2 = y + 2
Suitable Euclid’s axiom.
If equals are added to equals then the wholes are equal’.

Question 9.
Draw a diagram for the axiom ‘If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
Solution:
\(\overleftrightarrow{\mathrm{AB}}\) is a line and \(\overleftrightarrow{\mathrm{OC}}\) is ray stands on \(\overleftrightarrow{\mathrm{AB}}\).

∠BOC and ∠AOC are adjacent angles.
∴ ∠BOC + ∠AOC = 180°

9th Class Maths Introduction to Euclid’s Geometry 4 Marks Important Questions

Question 1.
Rohan’s maid has two children of same age. Both of them have equal number of dresses. Rohan on his birthday plans to give both of them same number of dresses. What can you say about the number of dresses each one of them will have after Rohan’s birthday? Which Euclid’s axiom is used to answer this question ? What value is Rohan j depicting by doing so ? Write one more i Euclid’s axiom.
Solution:
Let the two children of Rohan’s maid have the number of dresses as a and b respectively.
Then, according to the question,
a = b ………… (1)
Let Rohan plan to give both of them same number (K) of dresses on his birthday. ,
From Cl),
a + K = b + K
[If equals are added to equals, the wholes are equal. [Euclid’s Axiom (ii)]]
⇒ Number of dresses each one of them will have after Rohan’s birthday are equal.
Rohan is showing sympathy for his maid and affection for her two children.

One more Euclid’s axiom :
The whole is greater than the part.

Question 2.
Teacher held two sticks AB and CD of equal length in her hands and marked their mid points M and N respectively. She then asked the students whether AM is equal to ND or not. Arpita answered yes. Is Arpita correct ? State axiom of Euclid’s that support her answer. Which characteristics of Arpita you want to inculcate in your nature.
Solution:
AB = CD
⇒ \(\frac{1}{2}\) AB = \(\frac{1}{2}\) CD
[Things which are halves of the same things are equal to one another. [Euclid’s Axiom (vii)]]
⇒ AM = ND [∵ M is the mid-point of AB and N is the mid-point of CD]
⇒ Arpita is correct.
We want to inculcate the sharpness of brain in our nature, so that we may quickly and correctly reply to any query made by our teacher.

Question 3.
Residence Welfare Association planned to make a rectangular playground for children in their colony. What value are they exhibiting by doing so? What is the relation between all angles of a rectangle ? State the Euclid’s axiom used here. Also give any two other axioms.
Solution:
Residence Welfare Association feels concerned about the amusement of children. Children feel naturally highly pleased by playing together in a play-ground It creates sportsman spirit in their nature. Also, it keeps them physically fit and mentally sound. They learn how to behave in a group. Their inner happiness is greatly enhanced when they get a chance to play together in a playground.

All angles of a rectangle are equal in measure because each angle has a measure of 90°.

(Things which are equal to the same thing are equal to one another. [Euclid’s Axiom (i)])
Two other axioms:
i) Things which coincide with one another are equal to one another.
ii) If equals are added to equals, the wholes are equal.

Question 4.
In the figure given below, a line n falls on lines l 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.

Question 5.
In the figure given below, if ∠1 = ∠3, ∠2 = ∠4 and ∠3 = ∠4, write the relation 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).

Question 6.
In the figure given below, we have BX = \(\frac{1}{2}\)AB, BY = \(\frac{1}{2}\)BC and AB = BC. Show that BX = BY.

Solution:
Given : BX = \(\frac{1}{2}\)AB
BY = \(\frac{1}{2}\)BC
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]
\(\frac{1}{2}\)AB = \(\frac{1}{2}\)BC
BX = BY
Hence proved.

Question 7.
Fill the blanks with suitable answers which are given below.
(deductive reasoning, Theorem, Axiom, 13, 3, 2)
i) The number of dimensions, a solid has _________ .
ii) Euclid divided his famous treatise “The Elements” into: _________ chapters.
iii) Greeks emphasized on _________ .
iv) _________ is needs a proof.
Solution:
i) The number of dimensions, a solid has 3 .
ii) Euclid divided his famous treatise “The Elements” into : 13 chapters.
iii) Greeks emphasized on Deductive Reasoning.
iv) Theorem is needs a proof.

AP 9th Class Maths Important Questions Chapter 5 Co-Ordinate Geometry

Question 1.
Write any point lies on \(\overline{\text { OY }}\) (Positive Y – axis) and any point lies on \(\overline{\text { OX }}\) (Negative X-axis).
Solution:
Point lies on \(\overline{\text { OY }}\) be (x = 0, y ≥ 0)
example : (0, 2) (0, 3).
Point lies on \(\overline{\text { OX }}\) be (x = 0, y ≤ 0)
example : (2, 0) (3, 0).

Question 2.
The position of (3, 4) and (4, 3) are not the same on graph. Why?
Solution:
Given points (3, 4) and (4, 3) having x and y coordinates are equal.

Question 3.
The points such as (0, x), (0, -x), (0, y) and (0, -y) lie on the same line. Name the line.
Solution:
That line is Y – axis.

Question 4.
The co-ordinates of a point M(4, – 3). What are the distances of the point M from axes?
Solution:
Given point = M(4, -3)
Distance from x -axis to the M is | -3 | units = 3
Distance from y – axis to the M is | 4 | units = 4

Question 5.
From the figure given below, write the co-ordinates of the points A, B, P and Q.

Solution:
A(5,3), B(-4, -3), P(2, -2), Q(-3, 4)

Question 6.
What is a cartesian plane? Explain with a diagram.
Solution:
1) To locate the exact position of a point on a number line we need only a single reference.
2) To describe the exact position of a point on a Cartesian plane we need two references.
3) The two perpendicular lines taken in any direction are referred to as co-ordinate axes.
4) The horizontal line is called X – axis.
5) The vertical line is called Y – axis.
6) The meeting point of the axes is called the origin.
7) The distance of a point from Y – axis is called the x co-ordinate or abscissa.
8) The distance of a point from X-axis is called the y co-ordinate or ordinate.
9) The co-ordinates of origin are (0, 0),
10 ) The co-ordinate plane is divided into four quadrants namely Q₁; Q₂, Q₃, Q₄ i.e., first, second, third and fourth quadrants respectively.
11) The signs of co-ordinates of a point are as follows.
Q₁(+, +); Q₂; (-, +)
Q₃(-, -); Q₄; (+, -)

The coordinates of P are (1, – 2).
The coordinates of Q are (3, 4).
The coordinates of R are (- 2, – 5).
The co-ordinates of S are (- 4, 3).

Question 7.
Without using graph sheet write the location of points given below.
(-1, 3), (2, 0), (-3, -1), (0, -6)
Solution:
Given points A(-1, 3), B(2, 0), C(-3, -1) and D(0, -6)

Question 8.
Plot the points A (2, 2) B (6, 2) C (8, 5) and D (4, 5) in a graph sheet. Join all the points to make it a parallelogram. Find its area.
Solution:

Area of parallelogram = AB × BC
AB = | x₂ – x₁ | = | 6 – 2 | = 4
BC = | y₂ – y₁ | = | 5 – 2 | = 3
∴ Area = 4 × 3 = 12 sq. units.

Question 9.
Read the following table and answer the following questions given below.

i) The point belongs to Q₃
ii) The abscissa of the point C
iii) The point lie on X – axis
iv) The coordinates of origin
v) The point satisfy x > 0, y < 0
vi) The point satisfy x – y = 1
vii) The position of point B
viii) The Quadrant contain (3,-2)
Solution:
i) D
ii) 3
iii) F, H
iv) 0,0
v) C (3, – 2)
vi) A (2, 1)
Vii) Positive Y-axis
viii) Q₄

Question 10.
i) Plot the points 0(0, 0), A(0, 3), B(4, 3), D(4, 0) on a graph sheet.
ii) Join the points to form quadrilateral OABD. Identify the type of quadrilateral and Justify your answer.
iii) Find its area.
Solution:
i) Given points are
O(0, 0), A(0, 3), B(4, 3), D(4, 0)

ii) OABD represents a rectangle.
iii) Area of a rectangle OABD
= l × b = OD × BD = 4 × 3 = 12 sq units.

Question 11.
i) Plot the points A(2, 3), B(6, 3), C(4, 7) on a graph sheet.
ii) Join the points to form triangle ABC. Identify the type of triangle and justify your answer.
iii) Find its area,
Solution:
i) given points are A(2, 3), B(6, 3), C(4, 7)

ii) ΔABC is formed, which is an Isosceles triangle.
CD = | 7 – 3 |= 4 units
AB = |16 – 2 | =4 units

iii) Area of ΔABC = \(\frac{1}{2}\) × b × h .
= \(\frac{1}{2}\) × AB × CD
= \(\frac{1}{2}\) × 4 × 4
= 2 × 4 = 8 sq units