Circles Chords and Tangents
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.
Line AB is a tangent to the circle at P . To calculate PB. Use Pythagoras theorem
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.
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
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?
Tangent AC = Tangent AB
Therefore AB = 6.928 AC = 6.928
construct any triangle PQR
bisect any two angles of the triangle to meet at O
drop a perpendicular from the point O to the side between the two bisected angles to meet the side at S
with radius OS draw the circle
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.
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.
Given angle a1
and angle b1
find angle a2
since the angles in alternate segment are equal. Angle b2
since angles on alternate segment are equal.
Construct any triangle ABC
construct� the perpendicular bisector of any two sides of the triangle to meet at O
With O as the centre with radius OA or OB or OC draw the circle
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.
construct any triangle ABC
produce any two sides of the triangle
Bisect the two external angles of the triangle to meet at O
drop a perpendicular from the point O to the side between the two
bisected angles to meet the side at N
with centre O and radius ON draw the circle
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
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
Repeat the same for sides AB and BC
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
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 ABCanimation
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
Below are some photographs of direct pulley system and indirect pulley system.
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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