Circles Chords and Tangents | Mathematics Form 3

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Circles Chords and Tangents - Mathematics Form 3








 




Tangents

Definition
A tangent to a circle is a line that touches a circle at only one point A B.


The tangent forms a right angle with the radius of the circle centre O at the point of contact on A P B.


Note:

Line AB is a tangent to the circle at P . To calculate PB. Use Pythagoras theorem


Example 1

The diagram above shows a boy pushing a bicycle wheel of radius 8cm using a stick. ( note, the stick acts as the tangent to the rim which is a circle).
The distance from the centre of the circle and the point of the stick the boy is holding is 1m. Find the length of the stick.

Solution


Using Pythagoras theorem Length of stick is


.


A circle can have two tangents meeting externally at a certain point. These tangents are usually i)equal. ii)they subtend equal angles at the centre iii)the line joining the centre of the circle to the external point bisects the angle between the tangent



Example 2

The diagram below a sweet with stick being OA. Given that the radius of the sweet is 4cm and the length of the stick OA is 8cm. Calculate the given tangents AB and AC?

Solution


Tangent AC = Tangent AB


Therefore AB = 6.928 AC = 6.928











INSCRIBED CIRCLE

construct any triangle PQR

animation

bisect any two angles of the triangle to meet at O

animation

drop a perpendicular from the point O to the side between the two bisected angles to meet the side at S

animation

with radius OS draw the circle

animation

Note:
The circle touches all the three internal sides of the triangle. Such a triangle is known as inscribed circle and the center is called in-centre.


ALTERNATE SEGMENT

Considering the diagram below, angle ABD is formed by tangent CBD and chord AB at the point of contact. Angle ABD is equal to the angle subtended by chord AB in the alternate segment.

That is angle ABD = angle BRA.

Note: angles subtended by the same chord in alternate segment are equal.

Example

Given angle a1
= 30o
and angle b1
= 70o
find angle a2
and b2


Solution

Angle a2
= a1
= 30o
since the angles in alternate segment are equal. Angle b2
= b1
= 70o
since angles on alternate segment are equal.


CIRCUMSCRIBED CIRCLE

Construct any triangle ABC

 animation
construct  the perpendicular bisector of any two sides of the triangle to meet at O
animation

With O as the centre with radius OA or OB or OC draw the circle

animation
Note :
The circle touches all the three vertices of the triangle i.e.  at A, B and C
Such a circle is known as circumscribed circle and the centre is known as circumcentre.


ESCRIBED CIRCLE

construct any triangle ABC

produce any two sides of the triangle
animation

Bisect the two external angles of the triangle to meet at O

animation

drop a perpendicular from the point O to the side between the two
bisected angles to meet the side at N

animation

with centre O and radius ON draw the circle

animation

Note:

This circle touches all the three sides of the triangle ABC externally such

a circle is called inscribed circle and the centre is called the ex- centre


ORTHOCENTRE OF A TRIANGLE

The orthocenter of a  triangle is the point of intersection of the altitude of all the three sides of the triangle
draw any triangle ABC

animation

we can draw the altitude of one side of the triangle and repeat the same for the other two sides using a setsquare and ruler draw a line from the vertex to meet is opposite side at 90o


animation


Repeat the same for sides AB and BC
animation

NOTE

The lines AL, BN and CM are known as the altitudes of the triangle the point O where these altitudes meet is called the orthocenter the triangle.


CENTROID OF A TRIANGLE

The centroid of a triangle is the point of intersection of the three medians of the sides of the triangle the median is obtains by bisecting each side of the triangle to meet its opposite vertex

draw any triangle ABC

animation

Bisect the side say AC of the triangle (the same as the median of AC) to get K and join K to the vertex B.


Repeat the same to the sides AB and BC

The lines or bisector AR,BK and QC are known as the medians of the triangle the point O where the medians meet is called the centroid of the
triangle.

APPLICATION


Below are some photographs of direct pulley system and indirect pulley system.



EXAMPLE


Direct pulley system Picture of a bicycle chain.


The chain in the bicycle can be considered as a tangent.

Given that the distance between the small circle of radius 16cm and the bigger circle of radius 30cm is 1m.Find the length of the belt around the two rims of the bicycle.




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