LemonmathSelf-study sheet

Grade 9 · Lesson 5.4

Relative positions of a line and a circle

Name: Date: Time spent: min

Goals

Find the relative position of a line and a circle by comparing $d$ with $R$ .
Recognise and prove that a line is a tangent to a circle.
Use the property of two tangents from a point.

Remember

$d<R$ : they intersect (chord of length $2\sqrt{R^2-d^2}$ ); $d=R$ : they touch; $d>R$ : they do not meet.
A tangent is perpendicular to the radius at the point of contact, and conversely.
Length of a tangent: $MA=\sqrt{OM^2-R^2}$ .
Two tangents from $M$ : $MA=MB$ , and $MO$ , $OM$ are angle bisectors.

Exercises

Exercise 1

Three lines are at distances $4$ cm, $6$ cm and $7.5$ cm from the centre of the circle $(O, 6\text{ cm})$ . Find the position of each line relative to the circle.

Exercise 2

The line $a$ is at distance $8$ cm from the centre of the circle $(O, 10\text{ cm})$ . Prove that $a$ cuts the circle at two points $A$ , $B$ and find $AB$ .

Exercise 3

From a point $M$ at distance $15$ cm from the centre $O$ , draw the tangent $MA$ to the circle $(O, 9\text{ cm})$ . Find $MA$ .

Exercise 4

The triangle $ABC$ is right-angled at $A$ , with $AB=3$ cm and $AC=4$ cm. Prove that $AB$ is a tangent to the circle $(C, CA)$ and that the point $B$ lies outside this circle.

Exercise 5

From the top $M$ of a tower $80$ m high, we look out to sea to the horizon at $A$ , where $MA$ touches the Earth. Take a cross-section of the Earth to be a circle with centre $O$ and radius $6400$ km. Find $MA$ (give your answer to the nearest kilometre).

Exercise 6

From a point $M$ with $OM=8$ cm, draw the two tangents $MA$ , $MB$ to the circle $(O, 4\text{ cm})$ . Find the angle $\angle AMB$ and the measure of the minor arc $AB$ . (Revise arcs and chords.)

Exercise 7

(2.0 marks) From a point $M$ outside the circle $(O, 3\text{ cm})$ , with $OM=5$ cm, draw the two tangents $MA$ , $MB$ ( $A$ , $B$ are the points of contact). Let $H$ be the point where $OM$ meets $AB$ .

a) Prove that the four points $M$ , $A$ , $O$ , $B$ lie on one circle.

b) Prove that $OM\perp AB$ at $H$ .

c) Find $MA$ and $AB$ .

Exercise 8 Challenge

From a point $M$ outside the circle $(O)$ , draw the two tangents $MA$ , $MB$ with $MA=10$ cm. The tangent at a point $C$ on the minor arc $AB$ meets $MA$ and $MB$ at $D$ and $E$ respectively. Find the perimeter of the triangle $MDE$ .

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, 7\text{ cm})

and a line

a

at distance

7

cm from

O

. Which of the following statements is true?

a

cuts

(O)

at two points.B.

a

touches

(O)

.C.

a

does not meet

(O)

.D.

a

passes through

O

Question 2. A line at distance

3

cm from the centre of the circle

(O, 5\text{ cm})

cuts the circle in a chord of length

4

cm.B.

8

cm.C.

2\sqrt{34}

cm.D.

2

cm.

Question 3. From a point

M

with

OM=13

cm, draw the tangent

MA

to the circle

(O, 5\text{ cm})

. The length

MA

\sqrt{194}

cm.B.

12

cm.C.

8

cm.D.

18

cm.

Question 4. The tangents at

A

and

B

to the circle

(O)

meet at

M

, and

\angle AMB=50^\circ

. The angle

\angle AOB

50^\circ

.B.

40^\circ

.C.

130^\circ

.D.

25^\circ

Part II. True or false

Question 1. From a point

M

with

OM=12

cm, draw the two tangents

MA

MB

to the circle

(O, 6\text{ cm})

. Decide whether each statement is true or false.

a) $MA=6\sqrt3$ cm.	T / F
b) $\angle AMO=30^\circ$ .	T / F
c) $\angle AMB=30^\circ$ .	T / F
d) $MA=MB$ .	T / F

Part III. Short answer

Question 1. A line at distance

4

cm from the centre

O

cuts the circle

(O)

in a chord of length

6

cm. Find the radius (in cm) of the circle.

Question 2. The two tangents

MA

MB

to the circle

(O)

have

MA=7.5

cm. The tangent at a point

C

on the minor arc

AB

meets

MA

and

MB

D

and

E

. Find the perimeter (in cm) of the triangle

MDE

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

Hints (only when stuck)

Exercise 1.

What you need: compare $d$ with $R$ .

Steps:

Compare each distance with $6$ .

Exercise 2.

What you need: if $d<R$ they intersect; the chord has length $2\sqrt{R^2-d^2}$ .

Steps:

Compare $8$ with $10$ , then find the chord.

Exercise 3.

What you need: $OA\perp MA$ , so the triangle $OAM$ is right-angled at $A$ .

Steps:

Use Pythagoras' theorem with hypotenuse $OM$ .

Exercise 4.

What you need: the test for a tangent.

Steps:

Show that $A$ is on $(C, CA)$ and $AB\perp CA$ .
Compare $CB$ with $4$ .

Exercise 5.

What you need: $OA\perp MA$ ; convert $80$ m $=0.08$ km.

Steps:

Compute $OM=6400+0.08$ (km).
Use Pythagoras' theorem in the triangle $OAM$ , right-angled at $A$ .

Exercise 6.

What you need: $\sin\angle AMO=\dfrac{OA}{OM}$ ; the angles of a quadrilateral add up to $360^\circ$ .

Steps:

Find $\angle AMO$ , then $\angle AMB$ , then $\angle AOB$ .

Exercise 7.

What you need: the median to the hypotenuse is half the hypotenuse; the property of the perpendicular bisector.

Steps:

Let $I$ be the midpoint of $OM$ and prove that $IA=IB=IO=IM$ .
Show that $OM$ is the perpendicular bisector of $AB$ .
Find $AH$ using the area of the triangle $OAM$ .

Exercise 8.

What you need: the property of two tangents from a point, used at $D$ and at $E$ .

Steps:

Show that $DC=DA$ and $EC=EB$ , then write the perimeter in terms of $MA$ , $MB$ .

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