# Lesson Notes By Weeks and Term - Senior Secondary School 2

TANGENTS FROM AN EXTERNAL POINT

SUBJECT: MATHEMATICS

CLASS:  SS 2

DATE:

TERM: 3rd TERM

REFERENCE BOOKS

• New General Mathematics SSS2 by M.F. Macrae etal.
• Essential Mathematics SSS2 by A.J.S. Oluwasanmi.
• Exam Focus Mathematics.

WEEK ONE

TOPIC: TANGENTS FROM AN EXTERNAL POINT

Theorem:

The tangents to a circle from an external point are equal.

Given: a point T outside a circle, centre O, TA and TB are tangents to the circle at A and B.

To prove:  |TA| = |TB|

Construction:  Join OA, OB and OT

In  âs OAT and OBT

OAT = OBT = 900 (radius    tangent)

|OT| =  |OT|    (common side)

âOAT = âOBT (RHS)

|TA| =  |TB|

Note that

Example:

1.In the figure below O is the centre of the circle and the TA and TB are tangents if 0, calculate     < TBX

In  âTAX

AXT = 900 (Symmetry)

TAX = 180 – (900 + 390) sum of angles of ()

1800 – 1290 = 510

TBX = 510 (symmetry)

OR

â ATB is an Isosceles  triangle

|AT| =|BT|    (tangents from external point)

0 (symmetry)

< ATB = 2(39)  =  780

2TBX = 1800 – 780 (sum of angle in a

2 TBX   =    1020

TBX= 1020

2

TBX  =  510

2.PQR are three points on a circle Centre O. The tangent at P and Q meet at T. If   < PTQ = 620 calculate PRQ.

Solution

Join OP and OQ

POQ = 3600 – (900 + 900 + 620) sum of angle in a quadrilateral)

POQ = 3600 – 2420

POQ = 1180

PRQ = 1180 =  590 (2x angle at circumference = angle at centre)

2

R + R1 = 1800 (opp. angles of  a cyclic quadrilateral )

R1 = 1800 – R

R1 = 1800 – 590

R1 = 1210

PRQ  =  590  or 1210

Evaluation

1. ABC are three points on a circle, centre O such that 0, the tangents at B and C meet at T. Calculate  < BTC.

GENERAL EVALUATION/REVISION QUESTIONS

1. AB is a chord and O is the centre of a circle. If AOB = 780 calculate the obtuse angle between AB and the tangent B.

1 The dimension of a cuboid metal is 24cm by 21cm by 10cm, if the cuboid is melted and used in making a cylinder whose base radius is 15cm find the height of the cylinder.

2 The volume of a cylinder is 3600cm3 and its radius is 10cm calculate its

(a) curve surface area

(b) total surface area

Essential Mathematics, pages149-151, numbers 11-20.

WEEKEND ASSIGNMENT

Use the diagram below to answer the questions.

1.If  < ATO  =  360 ,calculate < ABO.

(a) 36   (b) 720      (c) 180     (d) 440

2.If 0, calculate   < AOT   (a) 1140     (b) 570    (c) 33   (d) 1230

3.If< BTO = 440, calculate 0   (b) 440    (c) 460   (d) 920

4.If   |AB| =  18cm and  |TB| = 15cm, calculate |TX|

(a) 180     (b) 330     (c) 780     (d) 120

5.If < AOT = 470, calculate ABO   (a) 470     (b) 940      (c) 1330       (d) 430

THEORY

1.O is the centre of a circle and two tangents from a point T touch the centre at A and B. BT is produced to C. If 0.calculate < ATC.

2.AD is a diameter of a circle,AB is a chord and AT is a tangent. a) State  the size of  0,find  the  size  of DAB in terms of   x.