Basic Geometrical Ideas

(Based on a standard textbook figure with points O, B, C, D, E)

(a) Five points: O, B, C, D, E

(b) A line: DB (or line passing through D and B, extending in both directions)

(c) Four rays: OB, OC, OD, OE

(d) Five line segments: OB, OC, OD, OE, DE

(Four points on a line: A, B, C, D)

The twelve ways to name the line using two letters are:
AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC.

(a) Line containing point E: Line AE

(b) Line passing through A: Line AE (or the line through A and O)

(c) Line on which O lies: Line CO (or line through C and O)

(d) Two pairs of intersecting lines:
Pair 1: Lines AE and CO (intersecting at O)
Pair 2: Lines AE and EF (intersecting at E)

(a) Through one given point: Infinitely many lines can pass through a single point.

(b) Through two given points: Only one unique line can pass through two distinct points.

(a) Two rays OP and OQ sharing common point O:
Place point O in the centre. Draw an arrow going right labelled P, and another arrow going left labelled Q. Both originate from O.
Ray OP goes in one direction; Ray OQ goes in the other.

(b) Point P on segment AB:
Draw a straight line from A to B. Mark a point P between A and B on that line.
P is an interior point of segment AB: A —•——•——• B
with the middle bullet representing P.

(a) True — All these points lie on line MN.

(b) True — M, O, N are collinear points lying on the segment.

(c) True — M and N are the two endpoints of segment MN.

(d) False — The endpoints of segment OP are O and P, not O and N.

(e) False — M is not an endpoint of segment QO; Q and O are.

(f) False — M lies on the opposite side of O from P, so M is not on Ray OP.

(g) True — Ray OP starts at O and goes through P; Ray QP starts at Q and goes through P. They are different rays.

(h) False — Ray OM goes towards M from O; Ray ON goes towards N (the opposite direction). They are opposite rays.

(i) False — Ray OM is indeed opposite to Ray OP, as M and P lie on opposite sides of O.

(j) False — O is the initial point of Ray OP.

(k) True — N is the starting point (initial point) of both Ray NP and Ray NM.

(Based on standard NCERT diagrams)

(a) Squiggly line that does not meet at the ends → Open curve

(b) A shape like a closed loop → Closed curve

(c) A wavy line with two free ends → Open curve

(d) A closed irregular shape → Closed curve

(e) A closed boundary → Closed curve

(a) Open Curve: A wavy line that begins at one point and ends at another without meeting. Like the letter S or a zigzag that doesn't close.

(b) Closed Curve: A curve that starts and ends at the same point, enclosing a region. Like a circle, oval, or closed loop.

A polygon is a simple closed curve made entirely of straight line segments. Examples:
Triangle (3 sides), Quadrilateral (4 sides), Pentagon (5 sides), etc.
The region enclosed inside the polygon is its interior. The boundary lines form the polygon itself.

(a) Yes, it is a curve (any shape drawn continuously without lifting the pen is a curve).

(b) Yes, it is closed (its boundary meets at a single point, enclosing a region).

(Standard figure with points A, B, C, D forming a quadrilateral-like shape)

The angles are named by their vertex letter or three points:

∠DAB, ∠ABC, ∠BCD, ∠CDA (the four interior angles).

(a) Interior of ∠DOE: Point A (lies in the region between rays OD and OE)

(b) Exterior of ∠EOF: Point C (lies outside the region bounded by rays OE and OF)

(c) On ∠EOF: Points E, O, F (on the rays themselves or at the vertex)

(a) One point in common: Two angles that share only their vertex (e.g., ∠AOB and ∠COD sharing only O).

(b) Two points in common: Angles sharing a ray except one endpoint — sharing vertex O and one point on a common ray.

(c) Three points in common: Two angles sharing two points on common rays plus the vertex.

(d) Four points in common: Two angles overlapping so four specific points coincide.

(e) One ray in common: Two adjacent angles sharing one arm. e.g., ∠AOB and ∠BOC share Ray OB.

In triangle ABC, mark vertices A, B, C. Draw P inside the triangle and Q outside.

Point A is a vertex of the triangle. It is therefore on the boundary — neither in the interior nor in the exterior.

(Based on standard figure with points A, B, C, D)

(a) Three triangles: ΔABC, ΔABD, ΔACD

(b) Seven angles: ∠A, ∠B, ∠C, ∠D, ∠ABD, ∠ABС, ∠ACD

(c) Six line segments: AB, AC, AD, BC, BD, CD

(d) Two triangles with ∠B as common: ΔABC and ΔABD

Draw a quadrilateral with vertices P, Q, R, S. Connect P to R and Q to S — these are the diagonals.

Diagonals are: PR and QS.

Their meeting point is in the interior of the quadrilateral.

(a) Opposite sides: KL & MN; LM & NK

(b) Opposite angles: ∠K & ∠M; ∠L & ∠N

(c) Adjacent sides: KL & LM; LM & MN

(d) Adjacent angles: ∠K & ∠L; ∠L & ∠M

(Based on standard figure with centre O; points A, B, C, D, E, P, Q on/near circle)

(a) Centre: O

(b) Three radii: OA, OB, OC

(c) Diameter: AC (line through centre O)

(d) Chord: ED (line segment joining two points on the circle)

(e) Two interior points: O and P

(f) Exterior point: Q

(g) Sector: The pie-slice region bounded by two radii and the arc between them. (e.g., region OAB)

(h) Segment: The region bounded by the chord ED and the arc ED.

(a) Yes. A diameter is a chord that passes through the centre. Since it joins two points on the circle, it satisfies the definition of a chord. So every diameter is a chord.

(b) No. A chord does not need to pass through the centre. Only the specific chord that passes through the centre is a diameter. So not every chord is a diameter.

Instructions for labelling a circle diagram:

• (a) Mark a point at the middle of the circle and label it O (centre).
• (b) Draw a line from O to a point on the circle and label it r (radius).
• (c) Extend the radius through O to the other side and label the full line d (diameter).
• (d) Draw two radii forming a pie-slice — that wedge region is a sector.
• (e) Draw a chord and shade the smaller part between the chord and arc — that is a segment.
• (f) Any point inside the circle boundary is interior.
• (g) Any point outside the circle boundary is exterior.
• (h) Any curved portion of the circle boundary is an arc.

(a) Through one point: Infinite line segments can be started (though they need two endpoints).

(b) Through two points: Only one unique line segment.

(c) Through three non-collinear points: 3 line segments (AB, BC, CA form a triangle).

Line: Extends infinitely in both directions. Has no endpoints. e.g., ↔

Line Segment: Has two fixed endpoints. Has a definite length. e.g., A——B

Ray: Has one starting point (initial point) and extends infinitely in one direction. e.g., O——→

The points A, B, C, D are collinear (on the same line).

Line segments are: AB, BC, CD, AC, BD, AD. Total = 6 segments.

(a) True. By definition, a triangle has 3 vertices (A, B, C), 3 sides (AB, BC, CA), and 3 interior angles (∠A, ∠B, ∠C).

(b) False. For a concave (non-convex) quadrilateral, the diagonals may intersect outside the interior.

(c) True. The diameter passes through the centre, making it the longest possible chord in any circle.

A simple closed curve is a closed curve that does not cross itself. It divides the plane into two regions: interior and exterior.

Examples: Circle and Triangle.

(a) True. OA, OB, OC are all radii of the same circle, so they are equal.

(b) True. The diameter is the longest chord, so AB > CD.

(c) False. CD does not pass through the centre O, so it is a chord but not a diameter.

Number of diagonals = n(n−3) ÷ 2.

Examples:
Triangle (3 sides): 3(3−3) ÷ 2 = 0 diagonals.
Quadrilateral (4 sides): 4(4−3) ÷ 2 = 2 diagonals.
Pentagon (5 sides): 5(5−3) ÷ 2 = 5 diagonals.

Vertex: P

Arms: Ray PQ and Ray PR

Angle: ∠QPR (also written as ∠RPQ or simply ∠P)

Vertices: A, B, C

Sides (line segments): AB, BC, CA

Angles: ∠A (∠BAC), ∠B (∠ABC), ∠C (∠BCA)

Interior: The triangular region enclosed inside.

Exterior: The infinite region outside the triangle.

Chord: A line segment joining any two points on the circumference.

Diameter: A special chord that passes through the centre. It is the longest chord and equals twice the radius (d = 2r).

For a regular polygon with n sides: Interior angle = (n − 2) × 180° ÷ n.
For hexagon (n = 6): = (6 − 2) × 180 ÷ 6 = 4 × 30 = 120°.

An angle of 90° is called a Right Angle.

OA and OB are the arms of the angle. O is the vertex. The angle is ∠AOB = 90°.

(a) Two lines: 0 (parallel lines), 1 (intersecting), or infinite (same line).

(b) Two circles: 0 (no overlap), 1 (tangent/touching), or 2 (intersecting).

(c) Line and circle: 0 (line outside), 1 (tangent), or 2 (secant/chord).

Curve: Any continuous drawing; sides may be curved (e.g., circle, oval).

Polygon: A special closed curve where all sides are straight line segments (e.g., triangle, quadrilateral).

Every polygon is a closed curve, but not every curve is a polygon.

The perpendicular from centre to chord bisects the chord.
Half of AB = 6 ÷ 2 = 3 cm.
OA = radius = 5 cm.
Let distance OM = d, where M is midpoint of AB.
By Pythagoras: d² + 3² = 5²
d² = 25 − 9 = 16.
d = 4 cm.

(a) 3 sides → Triangle

(b) 4 sides → Quadrilateral

(c) 5 sides → Pentagon

(d) 6 sides → Hexagon

(e) 8 sides → Octagon

(f) 10 sides → Decagon

A chord divides a circle into two parts.

Each part is called a segment. The smaller part is the minor segment and the larger part is the major segment.

A quadrilateral can be divided into 2 triangles by drawing a diagonal. Each triangle has angle sum 180°.
Sum of interior angles of a quadrilateral = 2 × 180° = 360°.

(a) Square → Closed

(b) Letter C → Open

(c) Circle → Closed

(d) Letter S → Open

(e) Triangle → Closed

The diameter is always twice the radius:
Diameter = 2 × Radius   (d = 2r)
Equivalently: Radius = Diameter ÷ 2   (r = d/2)

Example: If radius = 7 cm, then diameter = 14 cm.