Circles – chords and tangents
MathematicsPlane GeometryAges 14–15
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Plane Geometry
Circle (O; R) with a chord AB and an external point M. Drag the chord's midpoint H to change the distance OH and see that OH ⟂ AB and AB = 2√(R² − OH²); drag M to see the two tangents MT₁ and MT₂ are equal with OT ⟂ MT; drag the point on the circle to change the radius. Once M moves inside the circle, no tangent exists.
Lesson: Relationship between diameter and chord; tangents to a circle; two tangents from an external point
What it shows
Several circle theorems appear together on circle (O; R). The perpendicular from the centre to a chord AB bisects it at H, so by Pythagoras AB = 2√(R² − OH²): the closer a chord is to the centre, the longer it is, and when OH = 0 it is a diameter. A tangent is perpendicular to the radius at the point of contact. From an external point M there are exactly two tangents, MT₁ and MT₂; they have the same length √(OM² − R²), and MO bisects angle T₁MT₂. Through a point on the circle there is one tangent, and through a point inside it there is none.
How to use
Set Radius R and Distance from centre to chord OH, or drag H, and compare the displayed length of AB with a prediction from Pythagoras. Move H to the centre to obtain a diameter. With Show tangents from point M switched on, drag M towards the circle and watch MT shorten, then drag it onto the circle and inside it. Drag K to change the radius and see how both results respond.
Parameters you can change
- Radius R 2–8 cm
- Distance from centre to chord OH 0–7.5 cm
- Show tangents from point M
- Show diameter perpendicular to chord
Questions to explore
- For R = 5 cm and OH = 3 cm, how long is chord AB?
- Why are the two tangents drawn from an external point M always equal in length?
- How many tangents pass through a point on the circle, and through a point inside it?
