EUCLIDEAN GEOMETRY: (±50 marks)
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1 ULIN GMTRY: (±50 marks) Grade theorems:. The line drawn from the centre of a circle perpendicular to a chord bisects the chord. 2. The perpendicular bisector of a chord passes through the centre of the circle. 3. The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle. (n the same side of the chord as the centre) 4. ngles subtended by a chord of the circle, on the same side of the chord, are equal. 5. The opposite angles of a cyclic quadrilateral are supplementary. 6. Two tangents drawn to a circle from the same point outside the circle are equal in length. 7. The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment. 8. The angle on circumference subtended by the diameter equals 90 o. 9. xterior angle of a cyclic quadrilateral equals to the opposite interior angle. 0. line from the centre of a circle to a tangent is perpendicular on tangent. Grade 2 theorems:. Proportionality theorem, (Midpoint) 2. Similarity theorem (quiangular triangles)
2 xercise In the diagram below, is the centre of the circle. F I J K L G H escribe the following and use the figure above to write an example of each: a) iameter b) Radius c) hord d) Segment e) Sector f) rc g) Secant h) Tangent The following are some forms of logic applicable in proof of theorems and riders: If a b and b c then a c If a b c and d b c then a b d b, so a d If a b c and b d then a d c
3 ccording to the PS document there are seven theorems to be proved. The converses, where they exist, should be known to solve riders. Proofs of converses will not be examined. Theorems. The line drawn from the centre of the circle perpendicular to the chord bisects the chord. Hints Identify the information that is given and mark it on the figure. 2. The perpendicular bisector of a chord passes through the centre of a circle. Write down what you aiming at, i.e. R.T.P: onstruction will lead you to congruency Identify the given information and draw the figure. = How will you know that is the centre of the circle? Which line segments should we prove equal? onstruction will help to prove
4 3. The angle subtended by an arc at the centre of the circle is double the size of the angle subtended by the same arc at the circumference. The following is important: Subtended by arc / chord Investigate angle subtended by a diameter. Isosceles triangles and exterior angle of a triangle. 4. ngles subtended by a chord of a circle, on the same side of the chord, are equal. What is R.T.P.? onstruction will lead to isosceles triangles and exterior angles will assist to prove the theorem This theorem is directly based on the previous theorem x x 2x Therefore it is important for learners to understand the previous theorem Learners can investigate angles subtended by equal chords.
5 5. The opposite angles of a cyclic quadrilateral are supplementary. Pre-knowledge: pposite Quadrilateral yclic quadrilateral Supplementary (sum ) R.T.P: is a cyclic quadrilateral onstruction joining and can assist the learners to recognise the angle at the centre and the angle on the circle. Using sum of angles of a quadrilateral (360 Ways of proving that a quadrilateral is a cyclic quadrilateral: ngles subtended by the same arc are equal, that is pposite angles of a quadrilateral are supplementary F xterior angle of a quadrilateral isequal to the opposite interior angle. G
6 6. Two tangents drawn to a circle from the same point outside the circle are equal in length. Revise the following: Tangents to a circle Radius tangent ongruency Radii Identify the given and draw a figure 7. The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment. What is R.T.P.?y joining, and, it can be proved that (RHS) (congruency) Revise theorems and axioms pertaining to: Tangent to a circle Identify segments and alternate segments ngle subtended by the diameter Sum of angles of a triangle iameter tangent raw a figure and identify the given information. F. onstr: raw diameter F and join F and then apply the concepts. When a theorem is stated, identify: Information given in the statement and underline key words used What is to be proved. Then you need to be able to draw the sketch with the given statements and be able to what should be shown as a proof.
7 xercise2. is the midpoint of the chord and with on the circle. If = 300mm, and =50mm, calculate the radius of the circle is the chord of the circle with centre and is 24cm long. is the midpoint of. cuts the circle at. alculate the value of x if = 8cm. = m 8 x 3. and are two chords of a circle with centre. M is on and N is on such that M and N. lso = 50mm, M = 40mm and N = 20mm. etermine the radius of the circle and the length of. 4. is the centre of the circle below, LKP is a straight line and ˆ 2x 4. etermine Ô2 and Mˆ in terms ofx. 4.2 etermine ˆK and ˆK 2 in terms of x. 4.3 etermine Kˆ Mˆ. What do you notice? 4.4 Write down your observation regarding the measure of ˆK 2 and Mˆ K L 2 2 P M
8 5. is the centre of the circle below.mpt is the tangent and P MT. etermine, with reasons, x, y and z z 3 2 M P T 6. is a cyclic quadrilateral. MK is a tangent touching the circle at. bisects. If and intersect at and M = 50 0, and = 0 0, calculate: K 0 0 M 50 0 K 7. P and P are tangents to the circle at and respectively. P is parallel to. 7. Prove that: a) = b) bisects P P 7.2 If = 40 0, determine: a) b)
9 8. In the diagram below, two circles have a common tangent T. PT is a tangent to the smaller circle. PQ, QRT and NR are straight lines.let Qˆ x 8. Name, with reasons, THR other angles equal to x. 8.2 Prove that PTR is a cyclic quadrilateral. 9. In the figure TP and TS are tangents to the given circle. R is a point on the circumference. Q is a point on PR suchthat Qˆ Pˆ. SQ is drawn. Let Pˆ x. Prove that: 9. TQ SR 9.2 QPTS is a cyclic quadrilateral 9.3 TQ bisects SQˆ P
10 0. In the figure below, is the centre of the circle and PT = PR. Let R ˆ y and ˆ x. 0. xpress x in terms of y. 0.2 If TQ = TR andx= 20 calculate the measure of: (a) y (b) Rˆ 2
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