Triangles and Circles
Triangles and Circles combines similarity, Pythagoras' theorem and the standard circle theorems: angles in the same segment, the semicircle angle, tangents and cyclic quadrilaterals.
Core rules
- Basic Proportionality Theorem: a line parallel to one side of a triangle divides the other two sides in the same ratio.
- If two triangles are similar, corresponding sides are proportional and corresponding angles are equal.
- Pythagoras: in a right triangle, hypotenuse² = sum of squares of the other two sides.
- Angle in a semicircle is 90°. Angles in the same segment are equal.
- A tangent is perpendicular to the radius at the point of contact. Tangents from an external point are equal in length.
- Opposite angles of a cyclic quadrilateral sum to 180°.
Worked example
A point P is 13 cm from the centre O of a circle of radius 5 cm. Find the length of the tangent from P. Since the radius OT is perpendicular to tangent PT, triangle OTP is right-angled at T. PT² = OP² − OT² = 169 − 25 = 144, so PT = 12 cm. The other tangent from P is also 12 cm.
Traps
The tangent length is not OP − radius; it comes from Pythagoras. In a cyclic quadrilateral, adjacent angles are not supplementary - only opposite pairs are. For similarity, write vertices in matching order so ratios are set up correctly. Do not use Pythagoras unless a right angle is given or proved.