LemonmathSelf-study sheet

Grade 9 · Lesson 5.1

Introducing circles

Name: Date: Time spent: min

Goals

State the definition of a circle, its radius and its diameter.
Decide whether a point lies inside, on or outside a circle.
Know that a circle has its centre as a centre of symmetry and infinitely many lines of symmetry.

Remember

The circle $(O, R)$ is made of the points exactly $R$ from $O$ ; a diameter has length $2R$ .
$M$ lies on, inside or outside $(O, R)$ when $OM=R$ , $OM<R$ or $OM>R$ .
The centre is a centre of symmetry; every line through the centre is a line of symmetry.

Exercises

Exercise 1

Consider the circle $(O, 4\text{ cm})$ and three points $A$ , $B$ , $C$ with $OA=3$ cm, $OB=4$ cm and $OC=4.5$ cm. Does each point lie inside, on or outside the circle?

Exercise 2

The square $ABCD$ has side $4$ cm. Prove that the four vertices of the square lie on one circle. Find the radius of this circle (give your answer to 2 decimal places).

Exercise 3

Consider the circle $(O, 5\text{ cm})$ and a point $A$ on it. Let $A'$ be the reflection of $A$ in the point $O$ . Prove that $A'$ lies on the circle and find $AA'$ .

Exercise 4

A loudspeaker at $O$ can be heard clearly within a radius of $30$ m. From $O$ , house $H_1$ is $18$ m east and then $24$ m north; house $H_2$ : $20$ m east, $25$ m north; house $H_3$ : $10$ m east, $27$ m north. Which house hears it clearly, which is exactly on the edge, and which is outside the range?

Exercise 5

Minh cuts a round pizza of radius $15$ cm along a straight line through the centre. Are the two pieces equal? Find the length of the cut.

Exercise 6

Consider the circle $(O, 6\text{ cm})$ and a point $A$ on it. Take a point $M$ such that $\angle AOM=90^\circ$ and $\angle OMA=30^\circ$ . Does $M$ lie inside or outside the circle? (Revise trigonometric ratios of acute angles.)

Exercise 7

(1.5 marks) The triangle $ABC$ is right-angled at $A$ , with $AB=6$ cm and $AC=8$ cm. Let $M$ be the midpoint of $BC$ .

a) Find $BC$ .

b) Prove that $A$ , $B$ , $C$ lie on a circle with centre $M$ . Find the radius of this circle.

c) Let $D$ be the reflection of $A$ in $M$ . Prove that $D$ lies on the circle from part b) and that $ABDC$ is a rectangle.

Exercise 8 Challenge

Consider the circle $(O, 5\text{ cm})$ and a point $M$ with $OM=3$ cm. The point $A$ moves round the circle. Find the greatest and the least value of $MA$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. Consider the circle

(O, 6\text{ cm})

and a point

M

with

OM=5

cm. Which of the following statements is true?

M

lies outside the circle.B.

M

lies on the circle.C.

M

lies inside the circle.D.

M

is the centre of the circle.

Question 2. The diameter of a circle with radius

3.5

cm is

3.5

cm.B.

7

cm.C.

1.75

cm.D.

12.25

cm.

Question 3. How many lines of symmetry does a circle have?

1

.B.

2

.C.

4

.D. Infinitely many.

Question 4. The radius of the circle through the four vertices of a rectangle with sides

5

cm and

12

cm is

13

cm.B.

6

cm.C.

8.5

cm.D.

6.5

cm.

Part II. True or false

Question 1. Consider the circle

(O, 5\text{ cm})

, a point

A

with

OA=5

cm, a point

B

with

OB=4

cm and the point

C

, the reflection of

A

O

. Decide whether each statement is true or false.

a) The point $A$ lies on the circle.	T / F
b) The point $B$ lies outside the circle.	T / F
c) $AC=10$ cm.	T / F
d) The point $C$ lies on the circle.	T / F

Part III. Short answer

Question 1. Find the radius (in cm) of the circle through the four vertices of a square with side

6

cm (give your answer to 1 decimal place).

Question 2. Consider the circle

(O, 10\text{ cm})

and a point

M

with

OM=6

cm. The line through

M

perpendicular to

OM

cuts the circle at

A

and

B

. Find

AB

(in cm).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: compare the distance from the centre with $R$ .

Steps:

Compare $OA$ , $OB$ , $OC$ with $4$ .

Exercise 2.

What you need: the diagonals of a square are equal and bisect each other.

Steps:

Find $AC$ by Pythagoras' theorem.
Deduce that $OA=OB=OC=OD$ .

Exercise 3.

What you need: $O$ is the midpoint of $AA'$ .

Steps:

Deduce that $OA'=OA$ , then find $AA'$ .

Exercise 4.

What you need: Pythagoras' theorem; compare with $R=30$ .

Steps:

Compute $OH_1$ , $OH_2$ , $OH_3$ and compare them with $30$ .

Exercise 5.

What you need: every line through the centre is a line of symmetry.

Steps:

Use symmetry; the cut is a diameter.

Exercise 6.

What you need: the tangent of an acute angle is the opposite side divided by the adjacent side.

Steps:

In the triangle $OMA$ , right-angled at $O$ , write $\tan\angle OMA$ .
Find $OM$ and compare it with $6$ .

Exercise 7.

What you need: the median to the hypotenuse is half the hypotenuse.

Steps:

Use Pythagoras' theorem, then show that $MA=MB=MC$ .
Look at the diagonals of $ABDC$ .

Exercise 8.

What you need: the triangle inequality $OA-OM<MA<OA+OM$ .

Steps:

Take $A$ not on the line $OM$ .
Look at the two points where $OM$ meets the circle.

Self-check

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