1. What are the rotational symmetries of a square? How many lines of reflection symmetry does it have? What about a regular pentagon? A regular hexagon?
2. What is the length of the longest chord in a circle of radius 5 units? Is there a smallest chord?
3. The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points?
(Hint: We know that any point that is equidistant from two given points A and B lies on the perpendicular bisector of AB. Does this make the perpendicular bisector the locus? For this, we have to show that all the points on the perpendicular bisector are equidistant from A and B.)
- 1. What are the rotational symmetries of a square? How many lines of reflection symmetry does it have? What about a regular pentagon? A regular hexagon?
- 2. What is the length of the longest chord in a circle of radius 5 units? Is there a smallest chord?
- 3. The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points?
Step-by-step solution
Idea: A regular polygon with n sides comes back to itself after a turn of 360° ÷ n (and its multiples) and has n lines of symmetry. A chord is longest when it passes through the centre. For a locus, we must check both ways: every point of the locus has the property, and every point with the property is on the locus.
1. What are the rotational symmetries of a square? How many lines of reflection symmetry does it have? What about a regular pentagon? A regular hexagon?
- Square: it looks the same after a turn of 360° ÷ 4 = 90° about its centre, and so after 90°, 180°, 270° and 360°. These 4 turns are its rotational symmetries (order 4).½ mark
- Lines of symmetry of the square: the 2 diagonals and the 2 lines joining midpoints of opposite sides: 4 lines.½ mark
- Regular pentagon: turns of 360° ÷ 5 = 72°, that is 72°, 144°, 216°, 288°, 360° (order 5). Each line from a vertex to the midpoint of the opposite side is a line of symmetry: 5 lines.½ mark
- Regular hexagon: turns of 360° ÷ 6 = 60°, that is 60°, 120°, 180°, 240°, 300°, 360° (order 6). Lines of symmetry: 3 through opposite vertices and 3 through midpoints of opposite sides: 6 lines.½ mark
2. What is the length of the longest chord in a circle of radius 5 units? Is there a smallest chord?
- For any chord AB with centre O, the triangle inequality gives AB ≤ OA + OB = 5 + 5 = 10, with equality only when O lies on AB. So the longest chord is a diameter = 2 × 5 = 10 units.½ mark
- Moving a chord away from the centre makes it shorter and shorter: 1 unit, 0.1 unit, 0.001 unit, … For any short chord there is an even shorter one nearer the edge. So there is no smallest chord (its length can get as close to 0 as we like, but a chord must have two different end points).½ mark
3. The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points?
- Let the points be A and B, M the midpoint of AB and l the perpendicular bisector of AB. (a) Every point P on l is equidistant from A and B: in ΔPMA and ΔPMB, AM = BM, ∠PMA = ∠PMB = 90° and PM is common, so ΔPMA ≅ ΔPMB (SAS) and PA = PB.1 mark
- (b) Every point P with PA = PB lies on l: in ΔPMA and ΔPMB, PA = PB, AM = BM and PM is common, so they are congruent (SSS). Then ∠PMA = ∠PMB, and as they add to 180°, each is 90°. So PM is perpendicular to AB at its midpoint: P is on l. (If P = M, it is already on l.)1 mark
- Both directions hold, so the locus of points equidistant from A and B is exactly the perpendicular bisector of AB.
Check: Part 2: a chord at distance d from the centre has length 2√(25 − d2). At d = 0 this is 10, the largest value; as d gets close to 5 the length gets close to 0 but never reaches a smallest positive value.
Answer to write in the exam
1.
Square: rotational symmetries 90°, 180°, 270°, 360° (360° ÷ 4 = 90°); lines of symmetry = 4
Regular pentagon: 72°, 144°, 216°, 288°, 360° (360° ÷ 5 = 72°); lines of symmetry = 5
Regular hexagon: 60°, 120°, 180°, 240°, 300°, 360° (360° ÷ 6 = 60°); lines of symmetry = 6
∴ Square 4, pentagon 5, hexagon 6 lines of symmetry
2.
For chord AB, AB ≤ OA + OB = 5 + 5 = 10 (triangle inequality); equality when O lies on AB
∴ Longest chord = diameter = 10 units
Chords farther from the centre are shorter; their length approaches 0 but is never 0.
∴ There is no smallest chord.
3.
M = midpoint of AB, l = perpendicular bisector of AB
P on l: ΔPMA ≅ ΔPMB (SAS: AM = BM, ∠PMA = ∠PMB = 90°, PM common) ⇒ PA = PB
PA = PB: ΔPMA ≅ ΔPMB (SSS) ⇒ ∠PMA = ∠PMB = 90° (linear pair) ⇒ P on l
∴ Locus of points equidistant from A and B = perpendicular bisector of AB
Common mistakes that cost marks
- Counting only the diagonals of a square (2 lines) and forgetting the 2 lines through midpoints of opposite sides.
- Saying the smallest chord is “0”. A chord joins two different points of the circle, so its length is never 0, yet there is no smallest one.
- For the locus, proving only one direction (PA = PB ⇒ P on the bisector). A locus needs both directions, as the hint says.
How this can come in the exam
How many lines of symmetry does a regular octagon have?
- 4
- 6
- 8
- 16
Show answer
(C) 8
A regular polygon with n sides has n lines of symmetry: 4 through opposite vertices and 4 through midpoints of opposite sides, 8 in all.
The longest chord of a circle of radius 6.5 cm is
- 6.5 cm
- 13 cm
- 3.25 cm
- there is no longest chord
Show answer
(B) 13 cm
The longest chord is a diameter: 2 × 6.5 = 13 cm.
Try one yourself
What are the rotational symmetries of an equilateral triangle, and how many lines of symmetry does it have?
Show answer
Turns of 360° ÷ 3 = 120°: 120°, 240°, 360°. It has 3 lines of symmetry, each from a vertex to the midpoint of the opposite side.
More questions like this
- 1. How many circles pass through two points on a plane?
2. Are there circles of all possible radii passing through A and B? What is the radius of the smallest circle passing through A and B? What is the radius of the largest circle passing through A and B?
3. As you move away from segment AB along its perpendicular bisector, do the radii of the circles containing A and B increase or decrease?
4. As you go along the perpendicular bisector, will the circle drawn from that point through A and B appear more curved or less curved?
5. You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square? - How many circles can you draw through three distinct points A, B and C on a plane? Is there always at least one such circle? Not necessarily! What if A, B and C lie on a straight line, i.e., are collinear? Can you explain why, in this case, there is no circle through A, B and C?
- Let us assume that A, B and C are not collinear. Is there always a circle passing through A, B and C? Can there be more than one circle through A, B and C?
- Draw ΔABC with AB = 5 cm, ∠A = 70° and ∠B = 60°. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle?
- Draw ΔABC with AB = 5 cm, ∠A = 100°, AC = 4 cm. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle?