Extra Curriculum Support

SecA

1. The total number of propositions in the Euclid’s Elements is:

(a) 460 (b) 465

(c) 32 (d) 13

2. The number of dimensions, a surface has:

(a) 2 (b) 1

(c) 0 (d) 3

3. The edges of a surface are:

(a) Curves (b) None of these

(c) Point (d) Lines

4. Euclid stated that if equals are added to equals, the wholes are equal in the form of:

(a) A definition (b) An axiom

(c) A theorem (d) None of these

SecB

5. In how many lines two distinct planes can intersect ?

SecC

6. (i) AB = BC, M is the mid-point of AB and N is the mid-point of BC. Show that AM = NC.

(ii) BM = BN, M is the mid-point of AB and N is the mid-point of BC. Show that AB = BC.

Problem 1

1. In a construction project, a team is responsible for laying the foundation of a building. The team leader insists on using accurate measurements and adhering strictly to geometric principles, particularly ensuring that the angles and lengths conform to the design specifications.

Discuss how the principles of Euclid’s geometry, such as the exactness of measurements and the consistency of geometric constructions, reflect the values of integrity and fairness in real-life situations. Why is it important for professionals, like engineers and architects, to uphold these values? How can disregarding these principles lead to larger ethical issues?

Problem 2

2. Euclid’s dedication to discovering and organizing geometric truths exemplifies a lifelong commitment to learning and intellectual honesty. His work reminds us of the importance of seeking truth in all areas of life.

In what ways can the pursuit of truth, as demonstrated in Euclid’s geometric investigations, inspire you in your own education and personal growth? How does valuing truth and knowledge contribute to a more ethical and informed society? Provide an example where seeking the truth had a positive impact on your life or the lives of others.

Q1

1. Euclid’s fifth postulate states: "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 angles are less than two right angles."

Consider a scenario where two straight lines, when extended, appear to remain parallel. If these lines meet at some extremely distant point, what might this imply about the nature of the angles formed by a transversal cutting these lines? Can this situation arise in a non-Euclidean geometry? How does Euclid's fifth postulate change in such a geometry?

Q2

2. Euclid’s geometry is largely based on the notion of flat, two-dimensional space. Consider how Euclidean concepts might be generalized to three dimensions or even higher dimensions.

Choose one of Euclid’s postulates (other than the fifth) and generalize it to three dimensions. For instance, how would Euclid’s first postulate (a straight line can be drawn from any point to any point) be expressed in three-dimensional space? Discuss any new complexities that arise when moving from two to three dimensions, particularly in the context of defining and understanding geometric objects.

Q3

3. A surveyor needs to determine the distance between two points, A and B, located on opposite banks of a river, without directly measuring across the river. The surveyor constructs a right-angled triangle on one bank, with one leg along the bank and the other leg perpendicular to the river.

Using Euclid’s axioms, explain how the surveyor can indirectly calculate the distance between points A and B. Consider the implications of Euclid's first postulate (a straight line can be drawn from any point to any point) and the use of similar triangles in your explanation.

Questions

1. Euclid’s second axiom is:

(A) The things which are equal to the same thing are equal to one another.

(B) If equals be added to equals, the wholes are equal.

(C) If equals be subtracted from equals, the remainders are equals.

(D) Things which coincide with one another are equal to one another.

2. Euclid’s fifth postulate is:

(A) The whole is greater than the part.

(B) A circle may be described with any centre and any radius.

(C) All right angles are equal to one another.

(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.

3. The things which are double of the same thing are:

(A) equal

(B) unequal

(C) halves of the same thing

(D) double of the same thing

4. Axioms are assumed:

(A) universal truths in all branches of mathematics

(B) universal truths specific to geometry

(C) theorems

(D) definitions

5. John is of the same age as Mohan. Ram is also of the same age as Mohan. State the Euclid’s axiom that illustrates the relative ages of John and Ram:

(A) First Axiom (B) Second Axiom

(C) Third Axiom (D) Fourth Axiom

6. If a straight line falling on two straight lines makes the interior angles on the same side of it, whose sum is 120°, then the two straight lines, if produced indefinitely, meet on the side on which the sum of angles is:

(A) less than 120° (B) greater than 120°

(C) is equal to 120° (D) greater than 180°

Questions

1. Write whether the following statements are True or False? Justify your answer :

(1) Euclidean geometry is valid only for curved surfaces.

(2) The boundaries of the solids are curves.

(3) The edges of a surface are curves.

(4) The things which are double of the same thing are equal to one another.

(5) If a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C.

(6) The statements that are proved are called axioms.

(7) “For every line l and for every point P not lying on a given line l, there exists a unique line m passing through P and parallel to l ” is known as Playfair’s axiom.

(8) Two distinct intersecting lines cannot be parallel to the same line.

(9) Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.

2. Write whether the following statements are True or False? Justify your answer.

(i) Pyramid is a solid figure, the base of which is a triangle or square or some other polygon and its side faces are equilateral triangles that converges to a point at the top.

(ii) In Vedic period, squares and circular shaped altars were used for household rituals, while altars whose shapes were combination of rectangles, triangles and trapeziums were used for public worship.

(iii) In geometry, we take a point, a line and a plane as undefined terms.

(iv) If the area of a triangle equals the area of a rectangle and the area of the rectangle equals that of a square, then the area of the triangle also equals the area of the square.

(v) Euclid’s fourth axiom says that everything equals itself.

(vi) The Euclidean geometry is valid only for figures in the plane.

Questions

1. Ram and Ravi have the same weight. If they each gain weight by 2 kg, how will their new weights be compared ?

2. Solve the equation a – 15 = 25 and state which axiom do you use here.

3. Read the following statement:

“A square is a polygon made up of four line segments, out of which, length of three line segments are equal to the length of fourth one and all its angles are right angles”.

Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all angles and sides of a square are equal?

4. Read the following statements which are taken as axioms :

(i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.

(ii) If a transversal intersect two parallel lines, then alternate interior angles are equal.

Is this system of axioms consistent? Justify your answer.

Q1

1. A construction team is working on a triangular plot of land. The sides of the triangle are given as 7 m, 8 m, and 9 m. The team is tasked with verifying whether they can construct a triangle using these measurements. They need to use Euclid’s postulates and axioms to justify the construction.

(a) According to Euclid's first postulate, what can be said about the possibility of constructing a triangle with sides 7 m, 8 m, and 9 m?

(b) Which Euclidean axiom can be used to verify if the sum of any two sides of this triangle is greater than the third side?

(c) Using Euclid’s axioms, how would you explain the uniqueness of the triangle that can be constructed with these side lengths?

(d) If a perpendicular is drawn from one vertex of the triangle to the opposite side, explain how Euclid’s postulates ensure the existence and uniqueness of this perpendicular.

Q2

2. In a city planning project, two streets are to be laid out such that they are parallel to each other. The surveyor claims that if two lines are intersected by a transversal, and the corresponding angles are equal, then the lines are parallel.

(a) Which of Euclid’s postulates or axioms justifies the surveyor’s claim about the parallel lines?

(b) If the streets are not exactly parallel and deviate slightly, how would this impact the corresponding angles formed by the transversal? Use Euclid's geometry to explain.

(c) Describe how Euclid’s fifth postulate (also known as the parallel postulate) applies to the layout of these streets.

(d) If an alternate interior angle is found to be unequal, what conclusion can be drawn about the lines, and how does this relate to Euclid's axioms?