Constructions (Circles)
340. The Tangent's Promise · always at right angles to the radius
OP is always perpendicular to the tangent at P.
A tangent to a circle is perpendicular to the radius at the point of contact, so OP ⊥ tangent at P. A tangent meets the circle at exactly one point.
What this lesson covers
Try to break it
Drag P around the circle. The tangent at P always meets the radius OP at exactly 90°. Try to tilt the tangent off 90°; the geometry won't allow it. If it tilted, the line would cut the circle a second time — and a tangent touches the circle at exactly one point.
How you build it
Draw a tangent at point P on a circle.
- Draw a circle with centre O of any radius.
- Mark any point P on the circumference of the circle.
- Join the centre O to point P with a straight line.
- Construct a line through P that is perpendicular to OP. This is your tangent!
The proof, step by step
Prove that the constructed tangent is perpendicular to the radius at the point of contact.
- By definition, a tangent to a circle touches it at exactly one point (the point of contact).
- The shortest distance from the centre of a circle to any tangent line is the radius drawn to the point of contact.
- In geometry, the shortest distance from a point to a line is along the perpendicular. Therefore, the radius must be perpendicular to the tangent at the point of contact.
Worked example
A tangent PQ is drawn to a circle with centre O at point P. If OP = 7 cm and OQ = 25 cm, find the length of PQ.
Since the radius OP is perpendicular to the tangent PQ at P, triangle OPQ is a right-angled triangle. Using Pythagoras theorem: PQ² = OQ² - OP² = 25² - 7² = 625 - 49 = 576. Thus, PQ = √576 = 24 cm.
- 18 cm
- 24 cm — correct
- 15 cm
- 21 cm