© Omega Education Unit
Problems generally fall into a few basic groups: viz; Introduction:- Circle Theorem
This topic is simply about remembering a set of rules, so even if you struggle with math, this topic is where you can “Pick up” marks quite easily during the CXC exam.
• • •
Problems with No tangent Problems with One tangent Problems with Two ta tangents
Problems with One Tangent If the circle has one tangent, then look for any Here are a few tips to solve the majority of these problems. opportunity to use the theorem which says that the angle Always observe the diagrams carefully and be on the look- between tangent and chord is equal to angle in the out for : alternate segment. 1. 2. 3. 4. 5. 6. 7. 8.
Paral aralllel lin lines - al alter ternate nate angl angles es,, (Z (Z an angles gles)) Isosc soscel eles es tria riangl ngle ( base base angl ngles are are equ equal al)) Tangents ( see below) Cycli clic qu quadri adrila late terral ( opp opp.. Angs Angs are are sup supp) p) Angl ngles at at the the cent enter = twic wice ang ang.. at at cir circum cum. 0 Angles in a semicircle = 90 angles in the same se segment ar are equal angle between radius and tange ngent = 900
Once you understand what to do with the information above, then you should see the solution in less than 1 minute, and be able to complete the problem in less than 5 minutes. Examples: Parallel lines - alternate angles, (Z angles) •
Notice that angle between Tangent Tangent FG and Chord HI is , so the angle in the 20 Alternate segment x = 20. Similarly y = 35 o
Problems with Two Tangents
Alternate angles are equal ( angle x at the elbows of the “Z “ Z” are the same) ************************************************************
You will need to remember the following: • • •
Angle between radius and tangent =
0
90
© Omega Education Unit - 876 469-2775 Email:
[email protected] [email protected],, website: www.cxcdirect.schools.officelive.com
•
9
© Omega Education Unit
In all other cases, tangent or no tangent, look out for the following
Construction using using Ruler and Compass only Tips: on Constructing angles: All the angles that you will be asked to construct using compass only, can be completed once you know three (3) basic things. 0
1.
How to construct a
90
2.
How to construct a
60
3.
How to bisect an angle
angle
0
All other angles are derived from the above: Terms you need to know: •
•
Angles at center (100) = twice angle at circumference (y=50) Isosceles triangles, are formed whenever one point of a triangle is at the center of the circle and the other two points touch the circumference ( Triangle OJR). This means that the two base angles (x) are equal.
•
Perpendicular – at right angles
•
Bisect – divide into two equal parts Perpendicular bisector – a line which is at right angles to another line and which passes through its mid point.
•
Example:
Explain how you would construct the following angles using ruler and compass only. 45
0
;
135
0
;
30
0
165
0
0
;
,
120
0
157.5
Solution: •
Angles in the same segment (JR) are equal
• Angle in a semicircle
© Omega Education Unit - 876 469-2775 Email:
[email protected], website: www.cxcdirect.schools.officelive.com
1)
0
is found by first constructing a 900 then bisecting ( divide equally) the 900 . 45
0
2)
135
3)
30
4)
165
5)
120
0
= =½
0
+
90
45
0
0
(construct
0
+
30
0
-
60
60
0
=
135
0
=
180
6)
22.5 = 1/2(45)
7)
157.5 = 180 – 22.5
and
0
60
and then bisect)
0
0
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