Maths- Pythagoras, Angle facts and Circle Theorems

Question 1

Pythagoras

Answer 1

In any right-angled triangle, the square of the longest side is the sum of the squares of the other two sides. This can be written in the formula:
a2 + b2 = c2

Question 2

There are 360° in a complete turn. So in half a turn there are 180°, and in a quarter of a turn 90°.

Answer 2

We sometimes call a quarter of a turn a right angle, and mark it with a square.
We can use these facts to solve simple problems. Here are some examples:

Question 3

Different types of angle

Answer 3

Any angle that is between 180° and 360° is called a reflex angle.

Question 4

Any angle that is less than 90° is called an acute angle.

Answer 4

You also need to remember that the angles in a triangle add up to 180°, and that the angles in a quadrilateral add up to 360°.

Question 5

Any angle which is between 90° and 180° is called an obtuse angle.

Answer 5

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Question 6

Circumference of a circle

Answer 6

The circumference is the length of the edge around a circle.
For any circle, the circumference is:
3.141592… × the diameter.
Or in symbols: C = (3.141592…)d
This is true for all circles and so 3.141592… is therefore a special, unique number, and we represent it with the Greek letter π.(The symbol π is called ‘pi’ in English and is pronounced ‘pie’).
So we can write the formula for the circumference of a circle as: C = πd
However, the diameter is equal to 2 × radius, (2r), so we can also write this formula as:
C = 2πr
It does not matter which of these formulae you use. But you must be careful to use the correct length for the formula (the radius or diameter).

Question 7

Radius

Answer 7

A straight line from the center to the circumference of a circle or sphere

Question 8

Diameter

Answer 8

A straight line passing from side to side through the center of a body or figure

Question 9

Circumference

Answer 9

The edge of a circle

Question 10

Arc

Answer 10

A part of the circumference of a circle or other curve.

Question 11

Chord

Answer 11

A straight line joining the ends of an arc.

Question 12

Tangent

Answer 12

A tangent to a circle is a line which just touches the circle.

Question 13

Sector

Answer 13

Section of a circle (like a pizza slice)

Question 14

Segment

Answer 14

Area between the chord and the circumference

Question 15

The angle at the centre of a circle

Answer 15

One of the rules of geometry is that the angle subtended at the centre of a circle is double the size of the angle subtended at the edge from the same two points.

Question 16

Angles in the same segment are equal

Answer 16

Angles in the same segment are equal

Question 17

Angles in a semicircle are 90°

Answer 17

Angles in a semicircle are 90°

Question 18

Opposite angles in a cyclic quadrilateral add up to 180°

Answer 18

A cyclic quadrilateral is a quadrilateral whose vertices all touch the circumference of a circle. The opposite angles add up to 180o.

Question 19

The angle between the tangent and radius is 90°

Answer 19

A tangent to a circle is a line which just touches the circle.

Remember:
A tangent is always at right angles to the radius where it touches the circle.

Question 20

Tangents from a point outside the circle are equal in length

Answer 20

Two tangents to a circle from a point are equal. Have a look at the line of symmetry. The two tangents fit together.

Question 21

Alternate segment theorem

Answer 21

The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment.
This is the circle property that is the most difficult to spot. Look out for a triangle with one of its vertices (corners) resting on the point of contact of the tangent.