Case I: Line Passing Through Centre of Circle and Bisecting Chord

Circle: Symmetric Property 1

Objective: You will learn a symmetric property of circles in this activity and how to apply it in real life.

Case I: Line passing through centre of circle and bisecting chord

Step 1: Draw a circle in the space below and label the centre of the circle O.

Step 2: Draw a chord (which is not a diameter) and label it AB.

Step 3: Find the midpoint of the chord AB and label the midpoint M.

Step 4: Draw a line from the centre O to the midpoint M.

Step 5: Measure ÐOMA and ÐOMB. Answer: ÐOMA = _________ ; ÐOMB = _________

Q1: What is the relationship between the line OM and the chord AB?

Q2: Observe your classmate’s worksheet. Is the relationship in Q1 true for his/her circle?

Case II: Line passing through centre of circle and perpendicular to chord

Step 1: Draw a circle in the space below and label the centre of the circle O.

Step 2: Draw a chord (which is not a diameter) and label it AB.

Step 3: Construct a line l passing through the centre O and perpendicular to the chord AB.

Step 4: Label the point of intersection of the line l and the chord AB as M.

Step 5: Measure the lengths of AM and BM. Answer: AM = _________ ; BM = __________

Q3: What is the relationship between the line l and the chord AB?

Q4: Observe your classmate’s worksheet. Is the relationship in Q3 true for his/her circle?

Case III: Perpendicular bisector of chord

Step 1: Draw a circle in the space below and label the centre of the circle O.

Step 2: Draw a chord (which is not a diameter) and label it AB.

Step 3: Construct the perpendicular bisector of the chord AB and label it l.

Q5: Does the perpendicular bisector of the chord pass through the centre O of the circle?

Q6: Observe your classmate’s worksheet. Is the relationship in Q5 true for his/her circle?

Summary

There are three conditions in Cases I, II and III:

(a) The line l (or OM) passes through the centre O of the circle.

(b) The line l (or OM) bisects the chord AB.

In each of the three cases, two of the three conditions are given and these imply the third condition is true. For example,

Case I: Given conditions (a) and (b), you have observed from Q1 and Q2 that condition (c) is true.

Complete the following sentences:

Case II: Given conditions _______ and _______, I have observed from Q3 and Q4 that condition _______ is true.

Case III: Given conditions _______ and _______, I have observed from Q5 and Q6 that condition _______ is true.

This is a symmetric property of circles because the circle is symmetrical about ___________ ________________

Real-Life Applications

Q7: Construct a circle that passes through the three given points A, B and C.

Q8. The diagram on the right shows the plan of a living room with a balcony (not drawn to scale). The living room is rectangular in shape (6 m by 4 m) and the balcony is an arc of a circle (see dimensions in diagram). Using a scale of 2 cm : 1 m, draw an accurate scale drawing of the living room with the balcony. The main question is how to draw the arc for the balcony.

Answer Keys

Q1: What is the relationship between the line OM and the chord AB?

The line OM is perpendicular to the chord AB.

(In fact, OM bisects AB as well and so OM is the perpendicular bisector of AB.)

Q3: What is the relationship between the line l and the chord AB?

The line l bisects the chord AB.

(In fact, l is perpendicular to AB as well and so l is the bisector perpendicular of AB.)

Q5: Does the perpendicular bisector of the chord pass through the centre O of the circle?

Yes

Summary

Case II: Given conditions __(a)__ and __(c)__, I have observed from Q3 and Q4 that condition __(b)__ is true.

Case III: Given conditions __(b)__ and __(c)__, I have observed from Q5 and Q6 that condition __(a)__ is true.

This is a symmetric property of circles because the circle is symmetrical about the line l (or OM).

Real-Life Applications

Q7: Construct perpendicular bisectors of AB and BC. The intersection will give the centre O of circle. The radius is OA (or OB or OC).

Q8. Label the balcony ABC as shown. Construct perpendicular bisector of AB and measure 1 m from AB to get the point C. Construct perpendicular bisector of AC (or BC). The intersection of the two perpendicular bisectors will give the centre O of circle. Radius of circle = OA (or OB or OC).