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State Board · Class 9 · Mathematics Part 2

Lesson 1: Basic Concepts in Geometry

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी

Origin of geometry and basic concepts

From daily life

After the river Nile flooded, farmers in Egypt had to measure their fields again to mark the borders. This everyday need for measuring land grew into the subject we call geometry.

Simple definition

Geometry (भूमिती) comes from the Greek words Geo (Earth) and Metria (measuring). Point, line and plane are basic concepts that are not defined.

Rule 1: Where geometry came from — Geo + Metria means "measuring the Earth". The Greek mathematician Thales, who went to Egypt, is said to have found the height of a pyramid from its shadow using similar triangles. In the Vedic period people built altars using geometry; the Shulba-sutra explains how to make shapes with a string. Aaryabhat, Varahamihir, Bramhagupta and Bhaskaracharya also contributed.

  • Easy : Easy: Geo means Earth and Metria means measuring.
  • Medium : Medium: A 1 m stick has a 2 m shadow, so a pyramid with a 100 m shadow is 50 m high (similar triangles).
  • Exam-type : Exam: Which book of the Vedic period explains building shapes with a string? The Shulba-sutra.

Rule 2: Undefined terms — Point, line and plane are not defined, just as numbers are not defined. Lines and planes are sets of points, and "line" always means a straight line. We understand them from models: the tip of a pin (point), a stretched thread (line), a flat table top (plane).

  • Easy : Easy: A pin tip is a model of a point.
  • Medium : Medium: A stretched thread is a model of a line.
  • Exam-type : Exam: A line is a set of points; so is a plane.
Remember

Point, line and plane are not defined. "Line" means a straight line.

Common mistake

Trying to define a point as "a tiny dot". A dot has size; a point has none.

Shortcut

Memory clue: "भू + मिती = पृथ्वीचे मोजमाप".

Spot the difference

TermModel
PointPin tip
LineStretched thread
PlaneTable top

Practice questions

Easy
  1. 1What does the word geometry mean?
  2. 2Which language does the word come from?
  3. 3Name the three undefined terms.
Medium
  1. 4Who is said to have found a pyramid's height from its shadow?
  2. 5Which Vedic text describes shapes made with a string?
  3. 6A line is a set of what?
Challenging
  1. 7A 2 m stick casts a 3 m shadow. A tree casts a 12 m shadow at that time. Its height?
  2. 8Why can't we define a point?
Show answers (try them yourself first)
  1. 1Measuring the Earth.
  2. 2Greek.
  3. 3Point, line, plane.
  4. 4Thales.
  5. 5Shulba-sutra.
  6. 6Points.
  7. 723 × 12 = 8 m.
  8. 8It is a basic concept; defining it needs other terms that themselves need defining.

Co-ordinates of points and distance

From daily life

On a ruler, the marks 3 and 8 are 8 − 3 = 5 cm apart. A number line works the same way: the distance is the difference of the co-ordinates, taken the right way round so it is never negative.

Simple definition

The number given to a point on a number line is its co-ordinate (निर्देशक). The distance d(A, B) = larger co-ordinate − smaller co-ordinate.

Rule 1: Distance on a number line — To find d(A, B), subtract the smaller co-ordinate from the larger. If A(2) and B(9), d(A, B) = 9 − 2 = 7. If C(−5) and D(−1), d(C, D) = −1 − (−5) = 4. If X(−7) and Y(5), d(X, Y) = 5 − (−7) = 12.

Points P(−3) and Q(4) on a number line with distance 7−5−4−3−2−10123456PQd(P, Q) = 4 − (−3) = 7Distance = larger co-ordinate − smaller co-ordinate

In the figure : In the figure the distance of two points on a number line is found by subtracting the smaller co-ordinate from the larger.

  • Easy : Easy: A(1), B(6): d(A, B) = 5.
  • Medium : Medium: P(−4), Q(3): d(P, Q) = 3 − (−4) = 7.
  • Exam-type : Exam: Find the points 6 units from the point with co-ordinate 2: 2 − 6 = −4 and 2 + 6 = 8.

Rule 2: Properties of distance — The distance between two different points is positive. The distance of a point from itself is 0. The distance is never negative, and d(A, B) = d(B, A). The distance d(A, B) is the length l(AB).

  • Easy : Easy: d(A, A) = 0.
  • Medium : Medium: d(P, Q) = d(Q, P) = 7.
  • Exam-type : Exam: Can d(A, B) = −3? No, distance is never negative.
Remember

Distance = larger − smaller. It is never negative.

Common mistake

Writing d(C, D) = −2 − 1 for C(−2), D(1). Subtract the smaller from the larger: 1 − (−2) = 3.

Shortcut

Memory clue: "मोठा − लहान = अंतर".

Spot the difference

PointsWorkingDistance
A(2), B(9)9 − 27
C(−5), D(−1)−1 − (−5)4
X(−7), Y(5)5 − (−7)12

Practice questions

Easy
  1. 1What is the co-ordinate of a point?
  2. 2How do we find d(A, B)?
  3. 3d(A, B) for A(3), B(10)?
Medium
  1. 4d(P, Q) for P(−2), Q(5)?
  2. 5What is d(A, A)?
  3. 6d(M, N) for M(−8), N(−3)?
Challenging
  1. 7Points 9 units from co-ordinate 4?
  2. 8Is d(A, B) always equal to d(B, A)? Why?
Show answers (try them yourself first)
  1. 1The number the point shows on the number line.
  2. 2Larger co-ordinate minus smaller.
  3. 37.
  4. 47.
  5. 50.
  6. 6−3 − (−8) = 5.
  7. 7−5 and 13.
  8. 8Yes, because we always subtract the smaller from the larger.

Betweenness of points

From daily life

Three towns lie on one straight road. If you go from the first to the second and then on to the third, the two parts of the trip add up to the whole trip. The second town is "between" the other two.

Simple definition

For three collinear points, Q is between P and R (P − Q − R) when d(P, Q) + d(Q, R) = d(P, R).

Rule 1: Betweenness — Of three distinct collinear points, exactly one lies between the other two. If d(P, Q) + d(Q, R) = d(P, R), then P − Q − R. Example: d(A, B) = 4, d(B, C) = 6, d(A, C) = 10 and 4 + 6 = 10, so B is between A and C.

Three collinear points A, B, C with distances 3, 4 and 7; 3 + 4 = 7 so B lies between A and CA (2)B (5)C (9)d(A, B) = 3d(B, C) = 4d(A, C) = 73 + 4 = 7 → B is between A and C (A − B − C)

In the figure : In the figure betweenness: 4 + 6 = 10, so B lies between A and C.

  • Easy : Easy: AB = 2, BC = 3, AC = 5, so B is between A and C.
  • Medium : Medium: PQ = 8, QR = 5, PR = 3: 3 + 5 = 8, so R is between P and Q.
  • Exam-type : Exam: Towns X, Y, Z: XY = 45 km, YZ = 60 km, XZ = 105 km. 45 + 60 = 105, so Y is between X and Z.

Rule 2: Not collinear, and points at a given distance — If no sum of two distances equals the third, the three points are not collinear. For distances 4, 5, 7: 4 + 5 = 9, 4 + 7 = 11, 5 + 7 = 12, none equals the third, so not collinear. Points 9 units from co-ordinate 4 are 4 − 9 = −5 and 4 + 9 = 13.

  • Easy : Easy: 1, 2, 4: 1 + 2 = 3 ≠ 4, and no other pair works, so not collinear.
  • Medium : Medium: 3, 4, 5 are not collinear since 3 + 4 = 7 ≠ 5.
  • Exam-type : Exam: Points 2 units from the point with co-ordinate −1: −3 and 1.
Remember

Add the two shorter distances. If they equal the longest, the points are collinear.

Common mistake

Adding the longest distance with another. Compare the sum of the two shorter with the longest.

Shortcut

Memory clue: "लहान + लहान = मोठे ⇒ मध्ये".

Spot the difference

DistancesCheckResult
4, 6, 104 + 6 = 10Collinear, B between
45, 60, 10545 + 60 = 105Y between
4, 5, 7No sum matchesNot collinear

Practice questions

Easy
  1. 1When is Q between P and R?
  2. 2How many of three collinear points lie between the others?
  3. 3AB = 3, BC = 5, AC = 8. Which point is between?
Medium
  1. 4Write "B is between A and C" in symbols.
  2. 5Are points with distances 2, 3, 6 collinear?
  3. 6Points 5 units from co-ordinate 0?
Challenging
  1. 7d(A, B) = 7, d(B, C) = 12, d(A, C) = 5. Which point is between?
  2. 8Cities P, Q, R: PQ = 90 km, QR = 40 km, PR = 130 km. Which city is in the middle?
Show answers (try them yourself first)
  1. 1When d(P, Q) + d(Q, R) = d(P, R).
  2. 2Exactly one.
  3. 3B.
  4. 4A − B − C.
  5. 5No, 2 + 3 = 5 ≠ 6.
  6. 6−5 and 5.
  7. 7A, since 5 + 7 = 12.
  8. 8Q, since 90 + 40 = 130.

Segments, rays, lines and congruent segments

From daily life

A cut piece of string has two ends (a segment). A torch beam starts at the bulb and goes on and on (a ray). A stretched road seems to go on both ways (a line). The middle of a ribbon folded in half is its midpoint.

Simple definition

A segment (रेषाखंड) has two end points, a ray (किरण) has one, a line (रेषा) has none. The midpoint (मध्यबिंदू) splits a segment into two congruent (एकरूप) parts.

Rule 1: Segment, ray and line — Segment AB is A, B and all points between them; its length is l(AB) = d(A, B). Ray AB is seg AB together with all points P such that A − B − P; A is its starting point. Line AB is ray AB together with its opposite ray.

A segment, a ray and a line drawn one below the otherSegment AB — two end points, has a lengthABRay AB — starts at A, goes on for everABLine AB — goes on for ever on both sidesAB

In the figure : In the figure segment, ray and line differ in how far they extend.

  • Easy : Easy: Seg AB has two end points.
  • Medium : Medium: Ray AB starts at A and passes through B.
  • Exam-type : Exam: Ray AB and ray BA are different; they start at different points.

Rule 2: Congruent segments and midpoint — Segments with equal lengths are congruent: seg AB ≅ seg CD when l(AB) = l(CD). Reflexive: seg AB ≅ seg AB. Symmetric: seg AB ≅ seg CD gives seg CD ≅ seg AB. Transitive: seg AB ≅ seg CD and seg CD ≅ seg EF give seg AB ≅ seg EF. If A − M − B and seg AM ≅ seg MB, M is the midpoint; every segment has one and only one midpoint.

Segment AB of 6 cm with midpoint M, AM and MB are 3 cm eachAMB3 cm3 cmseg AM ≅ seg MB, so M is the midpoint of seg AB (AB = 6 cm)

In the figure : In the figure the midpoint M divides a segment into two congruent parts.

  • Easy : Easy: l(AB) = 10 and M is the midpoint, so l(AM) = 5.
  • Medium : Medium: l(PQ) = 14, so each half is 7.
  • Exam-type : Exam: A(−3), B(9): the midpoint M has co-ordinate 3, since d(A, M) = 3 − (−3) = 6 and d(M, B) = 9 − 3 = 6.

Rule 3: Distance of a point from a line — If seg CD ⊥ line AB and D is on the line, then l(CD) is the distance of the point C from line AB and D is the foot of the perpendicular.

  • Easy : Easy: A perpendicular from C meets the line at D; l(CD) = 4 means the distance is 4.
  • Medium : Medium: The shortest path from a point to a line is the perpendicular.
  • Exam-type : Exam: A slanted segment from C to the line is longer than the perpendicular CD.
Remember

Segment: 2 ends. Ray: 1 end. Line: no end. Midpoint is unique.

Common mistake

Saying ray AB = ray BA. Their starting points differ.

Shortcut

Memory clue: "खंड 2 टोके, किरण 1, रेषा 0".

Spot the difference

FigureEnd pointsLength
Segment2Measurable
Ray1Endless
Line0Endless

Practice questions

Easy
  1. 1How many end points does a segment have?
  2. 2What does l(AB) mean?
  3. 3When are two segments congruent?
Medium
  1. 4A segment is 18 cm long. Its midpoint divides it into?
  2. 5How many midpoints does a segment have?
  3. 6What is the foot of a perpendicular?
Challenging
  1. 7A(2), B(10). Find the co-ordinate of the midpoint.
  2. 8Prove: if seg AB ≅ seg CD and seg CD ≅ seg EF, then seg AB ≅ seg EF.
Show answers (try them yourself first)
  1. 1Two.
  2. 2Length of segment AB, i.e. d(A, B).
  3. 3When their lengths are equal.
  4. 4Two parts of 9 cm.
  5. 5Exactly one.
  6. 6The point where the perpendicular meets the line.
  7. 76 (4 from each end).
  8. 8l(AB) = l(CD) = l(EF), so the lengths of AB and EF are equal.

Conditional statements, converse and proofs

From daily life

"If it rains, the ground gets wet" is true. But "If the ground is wet, it has rained" may be false: someone may have spilt water. Turning a statement around does not keep its truth.

Simple definition

A conditional statement (सप्रतिबंध विधान) has the form "If … then …". Swapping the two parts gives the converse (विपरीत विधान).

Rule 1: Conditional statements and converse — The part after "If" is the antecedent (पूर्वांग) and the part after "then" is the consequent (उत्तरांग). Interchanging them gives the converse. "If a figure is a square then it is a rectangle" is true, but its converse "If a figure is a rectangle then it is a square" is false.

A conditional statement and its converseConditional statement and its converseIf it rains (antecedent),then the ground is wet (consequent).interchangeIf the ground is wet,then it rains. (converse)A converse need not be true (the ground may be wet for another reason).

In the figure : In the figure the antecedent follows If and the consequent follows then; interchanging them gives the converse.

  • Easy : Easy: In "If it rains then the road is wet", the antecedent is "it rains".
  • Medium : Medium: Converse of "If x = 2 then x + 1 = 3" is "If x + 1 = 3 then x = 2".
  • Exam-type : Exam: "If a number ends in 0 then it is divisible by 5" is true; its converse is false (15 is divisible by 5).

Rule 2: Postulates, theorems and proofs — Euclid gave postulates (स्वयंसिद्धके), self-evident statements accepted without proof, such as "infinitely many lines pass through a point" and "one and only one line passes through two points". A property proved logically is a theorem (प्रमेय); the logical argument is the proof (सिद्धता). In a theorem, the antecedent is the given and the consequent is what is to be proved.

  • Easy : Easy: "One line passes through two points" is a postulate.
  • Medium : Medium: In "If two lines intersect then their vertically opposite angles are equal", the given is "two lines intersect".
  • Exam-type : Exam: What is to be proved in the same statement? Vertically opposite angles are equal.

Rule 3: Direct and indirect proof — Direct proof: lines AB and CD meet at O. ∠AOC + ∠BOC = 180° and ∠BOC + ∠BOD = 180° (linear pairs), so ∠AOC = ∠BOD. Indirect proof: assume the consequent is false and reach a contradiction. To prove "If n2 is even then n is even": suppose n is odd, n = 2k + 1, so n2 = 4k2 + 4k + 1, which is odd. This contradicts n2 being even, so n is even.

  • Easy : Easy: ∠AOC + ∠BOC = 180° is a linear pair.
  • Medium : Medium: If ∠AOC = 70°, then ∠BOC = 110° and ∠BOD = 70°.
  • Exam-type : Exam: In the indirect proof, the first step is to assume that n is odd.
Remember

Converse: swap antecedent and consequent. A true statement can have a false converse.

Common mistake

Assuming the converse is also true. Check it separately.

Shortcut

Memory clue: "जर = पूर्वांग; तर = उत्तरांग; उलट = विपरीत".

Spot the difference

TermMeaning
AntecedentAfter "If"
ConsequentAfter "then"
PostulateAccepted without proof
TheoremProved logically

Practice questions

Easy
  1. 1What is the antecedent?
  2. 2What is the consequent?
  3. 3How do we get the converse?
Medium
  1. 4What is a postulate?
  2. 5What is a theorem?
  3. 6Write the converse of "If a number is divisible by 6, it is divisible by 3". Is it true?
Challenging
  1. 7In a direct proof of vertically opposite angles, which angle pairs are linear pairs?
  2. 8What is the key idea of an indirect proof?
Show answers (try them yourself first)
  1. 1The part after "If".
  2. 2The part after "then".
  3. 3Interchange the antecedent and consequent.
  4. 4A self-evident statement accepted without proof.
  5. 5A property proved logically.
  6. 6"If divisible by 3, then divisible by 6" is false (9).
  7. 7∠AOC with ∠BOC, and ∠BOC with ∠BOD.
  8. 8Assume the consequent is false and reach a contradiction.

Quick revision

  • Distance = larger co-ordinate − smaller co-ordinate; Q is between P and R if d(P, Q) + d(Q, R) = d(P, R).
  • Every segment has exactly one midpoint; congruent segments have equal lengths.
  • Converse: interchange the antecedent and the consequent; a theorem is a property proved logically.