How long is the touch
One right triangle measures how far a tangent runs.
- O is the centre, T is 13 units away.
- OP is the radius: 5 units, with a right angle at P.
- PT is the tangent: PT² = 13² − 5² = 144, so PT = 12.
The triangle the tangent hides
Let T be an outside point and O the centre. The tangent from T touches the circle at P. Join OP and OT. The tangent length is PT, the distance from T to P.
OP is perpendicular to the tangent at P. So triangle OPT is right-angled at P. OP is the radius. PT is the tangent, the side we want. OT, the whole distance to the centre, is the longest side. It is opposite the right angle, so it is the hypotenuse.
By Pythagoras: PT² = OT² − OP². Subtract the squares, then take the square root.
The pond, with numbers
O is the centre of a pond, radius 5 m. A walker stands at T, 13 m from O. Find PT by Pythagoras. PT² = 13² − 5² = 169 − 25 = 144. So PT = 12 m.
| Distance from centre | Tangent length |
|---|---|
| 5 m | 0 m — the point is on the rim |
| 10 m | 5√3 m |
| 13 m | 12 m |
At 10 m, PT² = 10² − 5² = 100 − 25 = 75. So PT = 5√3 m.
Slide the point. Read the tangent.
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