Notes · Unit 9
Geometry and Polygons
Fourteen short slides on reasoning and proof, similar figures, regular polygons, interior and exterior angles, perimeter and area, and loci.
Press Next (or the right arrow key) to move on. On most slides, each press reveals one more idea.
ReasoningSimilar figuresPolygonsExterior anglesLocus
1 · Reasoning
Inductive and deductive reasoning
Inductive
Specific → general
Spot a pattern in a few cases and guess a general rule. The conclusion is probable, not proved.
Deductive
General → specific
Apply a rule already known to be true to one case. The conclusion is certain.
Deductive example. All squares have four equal sides. ABCD is a square. So ABCD has four equal sides.
Inductive trap. 3, 5 and 7 are prime, so “all odd numbers are prime”? But 9 = 3 × 3. One counterexample disproves a conjecture, while many examples never prove it.
2 · Mathematical language
Statements, axioms and theorems
Statement. A sentence that is either true or false. x + 3 = 7 is an open sentence, because its truth depends on x.
Axiom (postulate) is assumed true without proof. A conjecture is believed true but not yet proved.
Theorem is proved from accepted statements. A corollary follows easily from a theorem. The converse of “if p then q” is “if q then p”.
Proof. A chain of logical steps. Each step must rest on an axiom, a definition or an earlier theorem, so the conclusion is guaranteed.
A converse may fail. “If a polygon is regular, all its sides are equal” is true. Its converse is false: a rhombus has equal sides but is not regular.
3 · Similar figures
Same shape, not necessarily same size
Figures are similar when they have the same shape but not the same size. Corresponding angles are equal and corresponding sides are in the same ratio.
Always similar: any two squares. Not always: two rectangles. For 2 × 3 and 4 × 9, the ratios 2/4 and 3/9 differ.
Triangles. If two triangles are similar, their corresponding sides are proportional. Equal corresponding angles are enough to make them similar.
Example. Triangle ABC has sides 3, 4, 5 cm and is similar to PQR with PQ = 6 cm. The scale factor is 6/3 = 2, so QR = 2 × 4 = 8 cm.
Congruent vs similar. Congruent triangles are similar (ratio 1), but similar triangles need not be congruent.
Parallel line. In equilateral triangle ABC, if DE ∥ BC, corresponding angles give ∠D = ∠E = 60°, so ADE is also equilateral.
4 · Similar figures and size
Squares for area, cubes for volume
Area
ratio of areas = (ratio of lengths)2
Volume
ratio of volumes = (ratio of lengths)3
Doubling. Double a square’s side and its area becomes 4 times. Double a cube’s edge and its volume becomes 8 times.
Model. Scale 1 : 20. A model roof of area 3 m² means an actual roof of 3 × 202 = 1200 m2.
Tank. A 40 L tank, and a similar tank 3 times as tall: 40 × 33 = 1080 L.
Pyramids. Lengths 14 cm and 10 cm, larger surface area 392 cm². Smaller: 392 × (10/14)2 = 392 × 25/49 = 200 cm2.
Cylinders. Radii 3 ft and 2 ft, smaller volume 40 ft³. Larger: 40 × (3/2)3 = 135 ft3.
If the volume ratio is 8 : 27, the length ratio is the cube root, 2 : 3.
5 · Polygons
Regular and irregular
A polygon is a closed figure made of straight segments. The fewest sides it can have is 3.
Regular: all sides equal and all interior angles equal. Examples: equilateral triangle, square.
Irregular: at least one side or one angle differs. A rectangle (not a square) has equal angles but unequal sides. A rhombus has equal sides but unequal angles. Neither is regular.
Perimeter. For a regular polygon with n sides of length s: P = n × s.
6 · Interior angles
Split the polygon into triangles
Sum = (n − 2) × 180°
Joining one vertex to the others makes n − 2 triangles, each with angle sum 180°. A quadrilateral gives 2 triangles, so its angle sum is 360°.
Diagonals. A polygon with n sides has n(n − 3)/2 diagonals. A hexagon has 6 × 3 / 2 = 9.
Each angle of a regular polygon = (n − 2) × 180° ÷ n. Nonagon: 7 × 180° ÷ 9 = 140°. Hexagon: 120°.
Sum for 12 sides. 10 × 180° = 1800°.
Missing angle. In a hexagon the angles are 90°, 155°, 112°, 140°, 98° and x. Their sum is 720°, so x = 720 − 595 = 125°.
Not every sum is possible. 9 right angles = 810°. Then n − 2 = 810 ÷ 180 = 4.5, which is not a whole number, so no polygon has that angle sum.
7 · Exterior angles
Always 360° in total
At each vertex, interior + exterior = 180°, because they form a linear pair.
Sum of exterior angles (one per vertex) = 360° for every polygon, whatever the number of sides.
Regular polygon: each exterior angle = 360°/n.
Find the sides. Exterior angle 120°: n = 360/120 = 3, an equilateral triangle. Exterior angle 45°: n = 8. Interior angle 156°: exterior 24°, so n = 15.
Twice as big. If the interior sum is twice the exterior sum: (n − 2) × 180° = 2 × 360°, so n − 2 = 4 and n = 6, a hexagon.
Unknown angles. Six exterior angles (x+4)°, (3x−4)°, (7x−3)°, (2x+3)°, (8x−1)°, (9x+1)° add to 360°: 30x = 360, so x = 12 and the angles are 16°, 32°, 81°, 27°, 95°, 109°.
Impossible. An exterior angle of 50° would give n = 360 ÷ 50 = 7.2 sides. A regular polygon cannot have that.
8 · Regular polygons in use
Perimeter and area
Apothem a is the perpendicular distance from the centre to a side. It is also the radius of the inscribed circle.
P = n × s and A = ½ P a
The polygon splits into n triangles of base s and height a, so A = n × ½sa = ½(ns)a.
The polygon splits into n triangles of base s and height a, so A = n × ½sa = ½(ns)a.
Water tank base. Regular pentagon, side 8 ft, apothem 7 ft. P = 40 ft, A = ½ × 40 × 7 = 140 ft2.
Fencing. Regular hexagonal plot, side 10 m, fencing Rs 160 per metre. P = 60 m, cost = 60 × 160 = Rs 9,600.
Square room. Perimeter 48 ft means side 12 ft. With height 10 ft, the four walls have area 4 × 12 × 10 = 480 ft2.
9 · Locus
A path that follows a rule
A locus is the set of all points that satisfy a given condition. The plural is loci. Every point on the locus satisfies the condition, and every point satisfying it lies on the locus.
Fixed distance from a point. The locus is a circle in a plane (a sphere in three-dimensional space).
Fixed distance a from a line. A pair of parallel lines, one on each side of the line, each at distance a.
Equidistant from two parallel lines. A parallel line halfway between them.
Equidistant from two points A and B. The perpendicular bisector of AB, so PA = PB for every point P on it.
10 · Locus applications
Bisectors and the circumcircle
Equidistant from two intersecting lines. A pair of lines that bisect the angles formed by the two lines. Every point on a bisector is equidistant from both lines.
Three non-collinear points A, B, C. The perpendicular bisectors of AB and BC meet at O, so OA = OB = OC. The circle through the three points is the circumcircle and O is the circumcentre.
Concentric circles. Points that stay at fixed distances from the same point O trace circles with the same centre.
Two isosceles triangles on one base. XAB and YAB share base AB. Since XA = XB and YA = YB, both X and Y lie on the perpendicular bisector of AB, so line XY bisects AB at right angles.
11 · Common mistakes
Five traps to avoid
Doubling the side doubles the area. It makes it 4 times larger. Volume becomes 8 times.
Using (n − 2) × 180° for exterior angles. The exterior angles always add to 360°.
Calling a rhombus or rectangle regular. A regular polygon needs both equal sides and equal angles.
Treating a locus from two points as a circle. Equal distances from two points give a straight line, the perpendicular bisector.
Assuming a conjecture is proved by examples. One counterexample is enough to disprove it.
12 · Quick check
Try these, then tap to check
13 · Your turn
Practise Unit 9
Three timed MCQ tests with an explanation after every answer.
Practice questions are independently written and are not official FBISE papers.