Lesson 1: Symmetrical Properties of Circles

Video Lesson:

Competency:

At the end of this lesson, you will be able to:

• Discover the symmetrical properties of circles.
• Use the symmetrical properties of circles to solve related problems.
• Explain angle properties of circles in their own words.
• Apply angle properties of circles to solve related problems.

Key terms:

• Circle
• Centre
• Line of symmetry
• Chord 

Brainstorming Questions

1. What is a circle?
2. Draw a circle and indicate its center, radius and diameter.
3. What is line of symmetry?
4. How many lines of symmetry does an equilateral triangle have?

Answer

A circle is set of points in a plane each of which is equidistant from a fixed point in the plane. The fixed point is called the center of the circle and the constant distance is called its radius.

1. 𝑂 is the centre, 𝑅 is the radius and 𝐷 is the diameter.
2.  The line of symmetry can be defined as the axis or imaginary line that passes. Through the centre of the shape or object and divides it into identical halves.
3. Equilateral triangle has three lines of symmetry.

1.1. Introduction to Circles

Definition

A circle is the locus of points (set of points) in a plane each of which is equidistant from a fixed point in the plane.

• The fixed point is called the centre of the circle and
• The constant distance is called its radius.
• Thus, the circle is defined by its centre O and radius r.
• Recall that area of a circle, 𝐴 = 𝜋𝑟² and perimeter of the circle, 𝑃 = 2𝜋𝑟.
• Any diameter of a circle can be considered as a line of symmetry for the circle.
• An object can have zero lines of symmetry or it can have infinite lines of symmetry.

Example 1

Theorem:
The line segment joining the center of a circle to the midpoint of a chord is perpendicular to the chord.
Proof:
Given A circle with center 𝑂 and a chord (PQ) whose midpoint is 𝑀. We want to prove that ∠𝑂MP is a right angle. Draw the diameter (ST) through point M. Then, the circle is symmetric about (ST) and (PM) ≡ (QM) .

So, (ST) is perpendicular bisector of PQ and hence ∠𝑂MP is a right angle.

Example 2:

𝛥𝐴𝐵𝐶 is an equilateral triangle and circle O is its circumcircle. How many lines of symmetry does figure below have?

Solution:
An equilateral triangle has three lines of symmetry

Theorem:
The line segment drawn from the centre of a circle perpendicular to a chord bisects the chord.
Proof:
Given: A circle with centre 𝑂 and (ON) is drawn from centre 𝑂 perpendicular to the chord 𝐴𝐵 as shown in figure below.
We want to prove that (AN) ≡ (NB).
Join (OA) and (OB).

1. (OA) ≡ (OB) …radii of circle
2. m(∠𝐴𝑁𝑂) = 𝑚(∠𝐵𝑁𝑂) …both equal to 90°
3. (ON) ≡ (ON) …common side
4. ∆𝐴𝑂𝑁 ≡ ∆𝐵𝑂𝑁 …by RHS-criteria of congruency
5. (AN) ≡ (BN) …by step 4

Theorem:
Equal chords of a circle are equidistant from the center of the circle.
Proof:
Given: Chords (AB) and (CD) are equal in length. Construction: Join points 𝐴 and 𝐶 with center 𝑂 and drop perpendiculars from 𝑂 to the chords (AB) and (CD) (see figure). We want to prove: (OP) ≡ (OQ).

Characteristics of Chord

Theorem:
If the angles subtended by the chords of a circle are equal in measure, then the length of the chords are equal.

Proof:
From figure below, consider ∆AOB and ∆POQ.

Theorem:
Chords which are equal in length subtend equal angles at the Centre of the circle.

Proof:
From figure below, consider ∆AOB and ∆POQ. We want to prove ∠AOB ≡ ∠POQ.

Steps	Statement	Reason
1	AB ≡ PQ	Given
2	OA ≡ OB ≡ OP≡ OQ	Radii of the same circle
3	∆𝐴𝑂𝐵 ≡ ∆𝑃𝑂𝑄	SSS postulate of Congruence
4	m(∠AOB) = 𝑚(∠𝑃𝑂𝑄)	From step 3

Example 3

If the chord of a circle of radius 10 cm is 16 cm long. Find the distance of the chord from the centre.
Solution:

$$\begin{gathered} \text{Given:}AB=16\text{cm and}OB=10\text{cm} \\ \text{Thus}MB=8\text{cm; (}\frac{1}{2}\text{AB)} \\ O{M}^2+M{B}^2=O{B}^2 \\ OM=\sqrt{\left(OB\right){}^2–\left(MB\right){}^2} \\ =\sqrt{\left(10\right){}^2–8{}^2} \\ =\sqrt{(100–64)} \\ =\sqrt{\left(36\right)} \\ =6\text{cm} \end{gathered}$$

1.2 Angle Properties of Circles

1.2.1. Central Angles and Inscribed Angles

Activity

Define the following terms: chord, diameter, radius, tangent, secant, arc, major and minor arc.

Answer

1. The chord of a circle can be defined as the line segment joining any two points on the circumference of the circle.
2. The diameter is the longest chord in a circle.
3. The distance from the centre point to any endpoint on the circle is called the radius of a circle.
4. A tangent to a circle is a straight line which touches the circle at only one point.
5. A straight line that intersects a circle at two points is called a secant line.
6. An arc is a smooth curve joining two points. Consequently, on a circle, every pair of distinct points determines two arcs:

Two points lying on a circle actually define two arcs. The shorter is called the ‘minor arc‘ and the longer one is called the ‘major arc‘.

Remark:

1. A major arc is an arc connecting two endpoints on a circle and its measure is greater than180°or π. (See the figure below). Arc ACB (is major arc).

2. Minor arc is an arc connecting two endpoints on a circle and its measure is less than 180° or π. (See figure below) Arc ACB ( is minor arc. Usually denoted by two letters. So we call minor arc AC.

3. A major arc is usually referred to with three letters and a minor arc is usually referred to with only two letters.

4. A central angle is an angle formed by two radii with vertex at the center of the circle. (See figure below) ∠AOB is central angle.

5. An inscribed angle is an angle with vertex on the circle formed by two intersecting chords. (See figure below) ∠APB is inscribed angle.

Theorem:
If an inscribed and a central angle intercept the same arc, then the measure of an inscribed angle is half of the measure of a central angle.

Example 4

In each of the following figures, O is the centre of the circle. Calculate the measure of the angles marked x.

x = ½ (80°) = 40°

b.

x +<AOB=360° and <AOB=2 × 50° = 100°.

Hence, x=360°-100°=260°.

1.2.2. Measure of Central Angles and Inscribed Angles

Theorem:
Inscribed angles subtended by the same arc have the same measure.

Proof:

In Figure below, m(∠𝐴P𝐵) = ½ m(∠𝐴O𝐵)

m(∠𝐴Q𝐵) = ½ m(∠𝐴O𝐵)
Therefore, m(∠𝐴P𝐵) = m(∠𝐴Q𝐵).

Example 5

Calculate the marked angle in the following figure:

Solution:

$$\begin{gathered} \Delta OAC:\angle AOC=180°-(17°+17°)=146° \\ \Delta OAB:\angle AOB=180°-(32°+32°)=116° \\ x=360°-(146°+116°) \\ x=360°-262° \\ x=98° \end{gathered}$$

Theorem: Angle in a semicircle (Thales’ Theorem)
An angle inscribed in a semicircle is a right angle.

     

Example 6

Solution:

$$\begin{gathered} x=\frac{120°}{2} \\ x=60° \end{gathered}$$

1.2.3. Cyclic Quadrilateral

Definition:
A quadrilateral is said to be a cyclic quadrilateral if there is a circle passing through all its four vertices.

Theorem:

The sum of the measures of opposite angles in a cyclic quadrilateral is 180°.

Proof:
Given: A cyclic quadrilateral WXYZ is inscribed in a circle with centre 𝑂 as shown in figure below.
Construction: Join the vertices 𝑊 and 𝑌 with centre 𝑂.

We want to show: m(∠WXY) + m(∠ WZY) = 180°. Consider arc WXY and arc WZY

1. ∠WOY ≡ 2∠WZY (The angle subtended by same arc is half of the angle subtended at the center)
2. Reflex angle WOY ≡ 2arcWXY (the angle subtended by same arc is half of the angle subtended at the center).
3. m(∠WOY) + Reflex m(∠WOY) ≡ 360° (Using steps 1 and 2).
4. 2m(∠WZY) + 2m(∠WXY) = 360° (Using steps 1 and 2).
5. 2(m(∠WZY) + m(∠WXY)) = 360°(Why?).
6. m(∠WZY) + m(∠WXY) = 180° (Why?)

Example 7

If the measures of all four angles of a cyclic quadrilateral are given as (4𝑦 + 2), (𝑦 + 20), (5𝑦 – 2), and 7𝑦 respectively, find the value of 𝑦.

Solution:
The sum of all four angles of a cyclic quadrilateral is 360°. So, to find the value of 𝑦, we need to equate the sum of the given four angles to 360°.

(4𝑦 + 2) + (𝑦 + 20) + (5𝑦 – 2) + 7𝑦 = 360°
17𝑦 + 20 = 360°
17𝑦 = 340°.

Therefore, 𝑦 = 20°.