circle Theorems - MPotW

Memorising the circle theorems is a first step to solving problems involving circles.
Next we need to practice how to identidy the steps in solution.

Warm-up:

Circle Theorems

Find $∠ABC$

$A$ , $C$ and $D$ are points on a circle centred $O$ ;
$AB$ and $CB$ are tangents to the circle;
$AD$ and $CD$ are chords in the circle;
$∠ ADC = 115 °$ .

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Solution:

The reflex angle AOC is equal to $115 ° \times 2 = 230 °$ , because the angle subtended by an arc at the centre is twice the angle subtended at the circumference.

So the obtuse angle AOC is then 130° as angles around a point sum to 360°.

The quadrilateral ABCO has internal angles at A and C of 90°, because tangents and radii meet at 90°.

So, $∠ABC$ is $360 ° - 130 ° - 2 \times 90 ° = 50 °$

-oOo-

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The Mean of a set of numbers

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Next Three Terms of a Quadratic Sequence

Warm-up:

Find ∠ABC

Solution:

Find $∠ABC$