Geometry Coordinate geometry
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Students need to be familiar with all theorems on the course. They also need to be able to apply the theorems to solve problems. ●
Learn the proofs of theorems 11, 12 and 13.
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Learn all the constructions. Students need to understand the process involved. They can be asked to use these to solve problems. Pay particular attention to the details involved in constructing the centres of the triangles and their properties. For example, if asked to construct the location of a fire station that is equidistant from each of three towns, construct the circumcentre.
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Learn the following terms related to logic and deductive reasoning: theorem, proof, axiom, corollary, converse, implies, equivalent to, if and only if, proof by contradiction.
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Students need to be able to solve problems involving enlargements. Learn the properties of enlargements and their construction. ●
Solve problems involving slopes of lines. Make sure you have a full understanding of the different ways to get slope. Be able to find the slope of a line from its graph and from its equation. Remember that tan θ equals slope where θ is the angle that the line makes with the positive direction of the X-axis. Remember: If and have slopes and
l1 l2 m1 m2 respectively then:
∥l2 ⇔
l1 m1= m2
l1⊥l2 ⇔
m1×m 2=–1
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Students need to be able to use the formulae for coordinate geometry of the line: distance, midpoint, slope, area of a triangle, the perpendicular distance from a point to a line, the angle between two lines and dividing a line segment internally in a given ratio m:n. Remember: when using the formula for the area of a triangle, one of the vertices must be at (0,0).
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Students need to be familiar with the equations of the circle and find the centre and radius from an equation. You need to be able to find the equation of a circle given certain information. When asked to find the equation of a circle, you should draw a diagram to help visualise the problem.
When trying to find the equation of a circle, let the equation be
x2 + y 2+2 gx +2 fy + c =0, then find three equations in g,f and c and solve for them.
Example: Find the equations of the circles that contains the point (3,-1), has radius 5 and whose centre is on the line
2x +3 y=4.
Let the circle have equation s:
x2 + y 2+2 gx +2 fy + c =0
(3,-1) is on s so sub it in:
(3)2 + (-1)2 + 2g (3)+2 f(–1) + c =0
⇒6g
–2 f + c = –10 The centre (–g,–f ) is on the line
⇒
2x +3 y =4 –2g –3 f =4 Rearrange the equations: Rearrange –2g –3 f =4toget g=
Rearrange 6g –2 f + c = –10 to get c =–6 g +2 f– 10 Sub both of these into Therefore the circles have equations
2+13 y2 – 190x +92 y + 532 =0
13x and x2 + y 2+2 x –4 y –20=0