Circles and Constructions: Tangent Theorems and Compass Work
Circles and Constructions pairs tangent theorems with straight-edge-and-compass work: tangents to a circle, division of a segment, and similar triangles.
Tangent facts
A tangent meets the circle at exactly one point and is perpendicular to the radius at that point. From an external point exactly two tangents can be drawn and they are equal in length. If the centre-to-point distance is d and the radius r, each tangent has length √(d² − r²). The angle between the two tangents and the angle subtended at the centre are supplementary (sum 180°).
A chord of the larger of two concentric circles that touches the smaller circle is bisected at the point of contact, so half the chord = √(R² − r²).
Constructions to practise
- Tangent at a point on the circle: draw the radius and erect a perpendicular there.
- Tangents from an external point P: draw OP, bisect it at M, draw the circle with centre M and radius MO; where it cuts the given circle are the points of contact.
- Divide segment AB in the ratio 3 : 2: draw a ray from A, mark five equal arcs, join the fifth point to B, then draw a parallel through the third mark.
- Construct a triangle similar to a given triangle with a stated scale factor, using the same equal-arcs method.
Worked example
Two concentric circles have radii 5 cm and 3 cm. Find the chord of the larger circle that touches the smaller one. Half-chord = √(5² − 3²) = √16 = 4 cm, so the chord is 8 cm.
Traps that cost marks
- Draw and mark the radius to the point of contact as 90° in the justification.
- Construction arcs must be visible; freehand tangents score nothing.
- Justify the construction in words - the marks are for the reasoning, not only the figure.
- Do not substitute the diameter for the radius in √(d² − r²).