Circles: SPI Mathematics

Formulas (2)

• Length of a tangent from a right triangle · Tangent length
  $$t = \sqrt{d^2 - r^2} \qquad d = \sqrt{r^2 + t^2}$$
• Angles in the radius and tangent triangle · Angles at the point of contact
  $$\angle OQP = 90^\circ \qquad \angle QOP + \angle OPQ = 90^\circ \qquad \sin 30^\circ = \cos 60^\circ = \tfrac{1}{2}$$

Reference tables (1)

Watch out for (6)

• Giving the radius as the gap between parallel tangents
  → What a tangent is and how many there are
• Two tangents from a point on the circle
  → What a tangent is and how many there are
• Adding the squares when one is the hypotenuse
  → Length of a tangent from a right triangle
• Subtracting the lengths instead of their squares
  → Length of a tangent from a right triangle
• Putting the right angle at the wrong corner
  → Angles in the radius and tangent triangle
• Using sin where cos is needed
  → Angles in the radius and tangent triangle

Formulas (3)

• Two tangents from one point are equal · Equal tangents
  $$PA = PB \qquad AB + CD = AD + BC \text{ (circle touching all four sides)}$$
• The angle between two tangents and the angle at the centre · Tangents and radii
  $$\angle APB + \angle AOB = 180^\circ$$
• The circle inside a triangle · Radius of the incircle
  $$r = \frac{a + b - c}{2} \text{ (right triangle)} \qquad r = \frac{\text{Area}}{s}, \quad s = \frac{\text{perimeter}}{2}$$

Watch out for (6)

• Matching the wrong pair of pieces
  → Two tangents from one point are equal
• Thinking only the first point gives equal tangents
  → Two tangents from one point are equal
• Taking 90 minus the angle instead of 180 minus it
  → The angle between two tangents and the angle at the centre
• Forgetting the two right angles
  → The angle between two tangents and the angle at the centre
• Dividing the area by the whole perimeter
  → The circle inside a triangle
• Halving the hypotenuse
  → The circle inside a triangle

Formulas (3)

• The perpendicular from the centre bisects a chord · Chord length
  $$\text{chord} = 2\sqrt{r^2 - d^2} \qquad r^2 = d^2 + \left(\tfrac{\text{chord}}{2}\right)^2$$
• Angles made by an arc · Angles in a circle
  $$\angle AOB = 2\,\angle ACB \qquad \text{angle in a semicircle} = 90^\circ$$
• Opposite angles of a cyclic quadrilateral · Cyclic quadrilateral
  $$\angle A + \angle C = 180^\circ \qquad \angle B + \angle D = 180^\circ$$

Watch out for (6)

• Stopping at the half chord
  → The perpendicular from the centre bisects a chord
• Putting the whole chord in the right triangle
  → The perpendicular from the centre bisects a chord
• Halving when you should double
  → Angles made by an arc
• Using an angle from the other side of the chord
  → Angles made by an arc
• Adding neighbouring angles to 180
  → Opposite angles of a cyclic quadrilateral
• Matching the exterior angle with the angle next to it
  → Opposite angles of a cyclic quadrilateral