Circle Theorems 1

Circle Theorems for who?  IGCSE, GCSE, 

Approach

“Subtends” – what does that mean?
The grand theorem
Proof (not required for exams but not too difficult)
So what then follows: Angles on the same segment, Angle in a semi-circle, Cyclic quadrilaterals, Alternate segment theorem

Then Add Tangents to a Circle…
… and now we can answer questions  (Tips)

Approach:  Maths teachers like to start  easy Maths and then build it up to make it more complex.  But sometimes it can be best to start at the top.  We will start with the hard Circle Theorem and from then on it gets easier and easier.

Subtends:  First a difficult word that only Maths teachers and examiners use.  Students normally use simpler language.circle subtends

Maths Teachers say Students may say
Chord BC subtends an angle of 48º with A Chord BC makes an angle of 48º with A
Arc BC subtends an angle of 48º with A Or just:   angle BAC is 48º

The grand  theorem (Play with the geogebra applet) 

The Circle Theorem

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Graph of the Quadratic – Name the parts

For a Maths Studies exam question you need to know how to draw a quadratic graph and to know the names of key parts of the graph.

For SL you need to know the same things as pre-learning

A Quadratic Graph

Image

Quadratic means related to squares, and a quadratic graph has an x2 (x-squared) term in it.  This is a hint to remember what quadratic means.

The ∪ or ∩ shape is called a Parabola

The Vertex is the minimum point or minimum (sometimes the maximum point)

The y-intercept point is the point where the curve cuts the y-axis

The x-intercept points are the points where the curve cuts the x-axis

There may be 2, 1 or 0 x-intercept points.  2 if the minimum point is below the x-axis, 1 if it is on the s-axis, and 0 if it is above.

Intercept wording – be careful
Intercept = Intercept Point BUT Intercept ≠ Intercept Value

for the graph above:            y-intercept point = (0,5)
y-intercept value = 5
y-intercept    = (0,5)

IB exams are unusual because you must remember that intercept alone means the co-ordinates, e.g. (0,5).  This is true for Maths Studies, Maths SL/HL, and even for some other subjects like Economics HL.

The Intercept value is useful when we go back to the equation.
For y= x2 – 6x + 5               the y-intercept value is +5
and generally for    y= a x2 – bx + c               the y-intercept value is c
This is because on the y-axis x=0, and when x= 0 then y = 0 + 0 + c  = c

For y= x2 – 6x + 5               the x-intercept value are 1 and 5
When x=1 or x=5 then y =0
The x-intercept values (sometimes called the zeroes)  are the solutions of
the equation  0= x2 – 6x + 5

-shaped parabolas are drawn when the x2 term is minusMS Quadratic names 2

 ∪ shaped or ∩ shaped Parabola?MS Quadratic names 3 smileys