Non-Calc - Geometry and Measures

A TANGENT AND RADIUS MEET AT 90 DEGREES

• A tangent is a line that touches a single point of the circumference of a circle
• A tangent always makes an angle of exactly 90 degrees with the radius it meets at this point.

TWO RADII FORM AN ISOSCELES TRIANGLE

• Any two radii on the a circle form an isosceles triangle
• They don’t have the ticks on the side to show that they are the same length, but they are because the length of the radius remains constant everywhere around the circle.

THE PERPENDICULAR BISECTOR OF A CHORD PASSES THROUGH THE CENTRE OF A CIRCLE

• A chord is any line drawn across a circle
• No matter where the chord is on the circle, the line that cuts it exactly in half (at 90 degrees) will go through the centre of the circle.

THE ANGLE AT THE CENTRE OF THE CIRCLE IS TWICE THE ANGLE AT THE CIRCUMFERENCE

• The angle subtended (made at) the centre of the circle is exactly double the angle subtended at the circumference of the circle from the same two points
• Two ends of the same chord joint up to make this.

THE ANGLE IN A SEMI-CIRCLE IS ALWAYS 90 DEGREES

• A triangle drawn from the two ends of a diameter will always make an angle of 90 degrees where it hits the circumference of the circle
• This happens no matter where it is hit.

ANGLES IN THE SAME SEGMENT ARE EQUAL

• All triangles drawn from a chord will have the same angle where they touch the circumference
• Also, the two angles on opposite sides of the chord add up to 180 degrees.

OPPOSITE ANGLES IN A CYCLIC QUADRILATERAL ADD UP TO 180 DEGREES

• A cyclic quadrilateral is a 4-sided shape with every corner touching the circle
• Both pairs of opposite angles in one will add up to 180 degrees.

TANGENTS FROM THE SAME POINT ARE THE SAME LENGTH

• Two tangents drawn from an outside point are always equal in length
• They create two congruent right angled triangles.

THE ALTERNATE SEGMENT THEOREM

• The angle between a tangent and a chord always lies equal to ‘the angle in the opposite segment
• For example the angle made at the circumference by two lines drawn from ends of the chord.