\(MO \parallel NP\) in a circle with centre \(O\). \(M\hat{O}N = \text{60}°\) and \(O\hat{M}P = z\). Calculate the value of \(z\), giving reasons.
8.1 Revision
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8.2 Ratio and proportion

Chapter 8: Euclidean geometry
 Sketches are valuable and important tools. Encourage learners to draw accurate diagrams to solve problems.
 It is important to stress to learners that proportion gives no indication of actual length. It only indicates the ratio between lengths.
 To prove triangles are similar, we need to show that two angles (AAA) are equal OR three sides in proportion (SSS).
 Theorems are examinable and are often asked in examinations. It is also important that learners remember the correct construction required for each proof.
 Notation  emphasize to learners the importance of the correct ordering of letters, as this indicates which angles are equal and which sides are in the same proportion.
 If a length has to be calculated from a proportion, it helps to rewrite the proportion with the unknown length in the top left position.
8.1 Revision (EMCHY)
Types of triangles (EMCHZ)
Name 
Diagram 
Properties 
Scalene 
All sides and angles are different. 

Isosceles 
Two sides are equal in length. The angles opposite the equal sides are also equal. 

Equilateral 
All three sides are equal in length and all three angles are equal. 

Acuteangled 
Each of the three interior angles is less than \(\text{90}\)°. 

Obtuseangled 
One interior angle is greater than \(\text{90}\)°. 

Rightangled 
One interior angle is \(\text{90}\)°. 
Congruent triangles (EMCJ2)
Condition 
Diagram 
SSS (side, side, side) 
\(\triangle ABC \equiv \triangle EDF\) 
SAS (side, incl. angle, side) 
\(\triangle GHI \equiv \triangle JKL\) 
AAS (angle, angle, side) 
\(\triangle MNO \equiv \triangle PQR\) 
RHS (\(\text{90}\)°, hypotenuse, side) 
\(\triangle STU \equiv \triangle VWX\) 
Similar triangles (EMCJ3)
Condition 
Diagram 
AAA (angle, angle, angle) 
\(\hat{A} = \hat{D}, \enspace \hat{B} = \hat{E}, \enspace \hat{C} = \hat{F}\) \(\therefore \triangle ABC \enspace  \enspace \triangle DEF\) 
SSS (sides in prop.) 
\(\frac{MN}{RS} = \frac{ML}{RT} = \frac{NL}{ST}\) \(\therefore \triangle MNL \enspace  \enspace \triangle RST\) 
Circle geometry (EMCJ4)

If an arc subtends an angle at the centre of a circle and at the circumference, then the angle at the centre is twice the size of the angle at the circumference. 
Angles at the circumference subtended by arcs of equal length (or by the same arc) are equal. 
Cyclic quadrilaterals (EMCJ5)
If the four sides of a quadrilateral \(ABCD\) are the chords of a circle with centre \(O\), then:

Proving a quadrilateral is cyclic:
If \(\hat{A} + \hat{C} = \text{180}°\) or \(\hat{B} + \hat{D} = \text{90}´\), then \(ABCD\) is a cyclic quadrilateral. 
If \(\hat{A}_1 = \hat{C}\) or \(\hat{D}_1 = \hat{B}\), then \(ABCD\) is a cyclic quadrilateral. 
If \(\hat{A} = \hat{B}\) or \(\hat{C} = \hat{D}\), then \(ABCD\) is a cyclic quadrilateral. 
Tangents to a circle (EMCJ6)
A tangent is perpendicular to the radius (\(OT \perp ST\)), drawn to the point of contact with the circle. 
If \(AT\) and \(BT\) are tangents to a circle with centre \(O\), then:


The midpoint theorem (EMCJ7)
The line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side.
Given: \(AD = DB\) and \(AE = EC\), we can conclude that \(DE \parallel BC\) and \(DE = \frac{1}{2}BC\).
Revision
\(O\) is the centre of the circle with \(OC = \text{5}\text{ cm}\) and chord \(BC = \text{8}\text{ cm}\).
Determine the lengths of:
\(PQ\) is a diameter of the circle with centre \(O\). \(SQ\) bisects \(P\hat{Q}R\) and \(P\hat{Q}S = a\).
Write down two other angles that are also equal to \(a\).
Calculate \(P\hat{O}S\) in terms of \(a\), giving reasons.
Prove that \(OS\) is a perpendicular bisector of \(PR\).
\(BD\) is a diameter of the circle with centre \(O\). \(AB = AD\) and \(O\hat{C}D = \text{35}°\).
Calculate the value of the following angles, giving reasons:
\(O\) is the centre of the circle with diameter \(AB\). \(CD \perp AB\) at \(P\) and chord \(DE\) intersects \(AB\) at \(F\).
Prove the following:
\(QP\) in the circle with centre \(O\) is extended to \(T\) so that \(PR = PT\). Express \(m\) in terms of \(n\).
In the circle with centre \(O\), \(OR \perp QP\), \(QP = \text{30}\text{ mm}\) and \(RS = \text{9}\text{ mm}\). Determine the length of \(y\).
\(PQ\) is a diameter of the circle with centre \(O\). \(QP\) is extended to \(A\) and \(AC\) is a tangent to the circle. \(BA \perp AQ\) and \(BCQ\) is a straight line.
Prove the following:
\(BAPC\) is a cyclic quadrilateral
\(TA\) and \(TB\) are tangents to the circle with centre \(O\). \(C\) is a point on the circumference and \(A\hat{T}B = x\).
Express the following in terms of \(x\), giving reasons:
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8.2 Ratio and proportion
