Chords, Tangents and Secants
Lines meeting circles have power relationships.
About this topic
Several theorems describe lines meeting circles. A radius perpendicular to a chord bisects it, and equal chords are equally far from the centre. Tangents drawn to a circle from the same outside point are equal in length.
When chords cross inside a circle the products of their segments are equal, and similar power-of-a-point relationships hold for secants and tangents. These rules produce equations to find unknown lengths, and are used in design of gears, wheels and arches.
Worked example: two chords cross inside a circle, one cut into pieces 4 and 6, the other into 3 and x. By the intersecting chords rule, 4 × 6 = 3x, so x = 8. Two tangents from an external point P touch the circle at A and B, so PA = PB. If a radius is 5 and a chord is 8 long, the distance from the centre to the chord is √(25 − 16) = 3.
Mechanical designers use tangents in belts and pulleys, and road engineers use them to blend straight roads into curves. A common mistake is multiplying the whole chord lengths instead of the two segments of each chord. Another is forgetting that a tangent meets the radius at 90°, which is the key to many right triangle problems. For a tangent and secant from the same point, the tangent squared equals the whole secant times its external part. Draw the radius to the contact point.
Key ideas
- A tangent is perpendicular to the radius
- Tangent segments from one point are equal
- Intersecting chords: ab = cd
Quick check
Use cases: where Chords, Tangents and Secants helps
- Homework: review the key idea behind Chords, Tangents and Secants before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Chords, Tangents and Secants without notes.
- Visual learning: play with the interactive lab to see how Chords, Tangents and Secants changes when you adjust its inputs.
- Teaching a friend: explain Chords, Tangents and Secants in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Chords, Tangents and Secants?
Lines meeting circles have power relationships.
What will I learn in Chords, Tangents and Secants?
A tangent is perpendicular to the radius. Tangent segments from one point are equal. Intersecting chords: ab = cd.
Is Chords, Tangents and Secants part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Chords, Tangents and Secants lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Chords, Tangents and Secants?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
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