🔍 Grade 10 · Lesson 5 of 12
Tangent Sweeper: The Tangent-Radius and Alternate Segment Theorems
A tangent always meets the radius at a perfect right angle, and the angle it makes with a chord secretly equals an angle hiding on the far side of the circle.
💡
The big idea: A tangent line touches a circle at exactly one point and is always perpendicular to the radius drawn to that point. That single right angle unlocks Pythagorean calculations for tangent lengths, and combined with the alternate segment theorem, it also lets you find angles between a tangent and a chord without any extra measuring.
A line that just grazes the circle
A tangent is a straight line that touches a circle at exactly one point — the point of contact — without ever crossing into the interior. No matter where that point is on the circle, one thing is always true about the tangent there.
🔑
The tangent-radius theorem
A tangent line is always perpendicular to the radius drawn to the point of contact. That single 90° angle is the key that unlocks almost every tangent problem.
\[ PT = \sqrt{OP^2 - r^2} \]
If O is the center (radius r), P an external point, and T the point of tangency, triangle OTP has a right angle at T — so Pythagoras gives the tangent length PT.
🎮 Tangent Sweeper LIVE
A tangent meets the radius at 90 degrees — sweep the point and watch the right angle hold.
The alternate segment theorem
Now draw a chord from the point of tangency back into the circle. The angle between the tangent and that chord looks like it should need a protractor — but it doesn't. The alternate segment theorem says that angle always equals the inscribed angle standing on the far side of the chord, in the “alternate” segment of the circle.
✨
Tangent-chord angle = angle in the alternate segment
The angle between a tangent and a chord drawn from the point of contact equals the inscribed angle subtended by that same chord in the alternate segment — the region of the circle on the other side of the chord.
📝 Worked example: From external point P, a tangent touches a circle of radius 9 at T. If OP = 41, find the tangent length PT.
- Triangle OTP has a right angle at T (tangent-radius theorem), with hypotenuse OP.
- PT² = OP² − OT² = 41² − 9² = 1681 − 81 = 1600.
- PT = √1600.
✓ PT = <strong>40</strong> — a scaled-up 9–40–41 right triangle.
📝 Worked example: The angle between a tangent and a chord at the point of contact is 50°. Find the inscribed angle in the alternate segment.
- By the alternate segment theorem, the tangent-chord angle equals the inscribed angle on the far side of the chord.
- That inscribed angle is therefore also 50°.
✓ The angle in the alternate segment is <strong>50°</strong>.
⚠️
Alternate means the other side
Don't match the tangent-chord angle to an inscribed angle on the same side of the chord as the tangent-chord angle — that pairing is not necessarily equal. The theorem specifically pairs it with the angle in the segment on the opposite side of the chord.
✨
Two tangents from one point are always equal
If you draw both tangents from the same external point P, the two triangles OTP and OT'P are congruent (they share hypotenuse OP and both have a leg of length r meeting it at 90°). So the two tangent segments PT and PT' are always exactly equal in length.
Check your understanding
1. A tangent line touches a circle at point T. What angle does it make with the radius OT drawn to that point?
The tangent-radius theorem ensures a right angle at the point of contact, for every circle and every tangent.
2. From external point P, a tangent touches a circle of radius 9 at T. If OP = 41, what is the tangent length PT?
By Pythagoras on right triangle OTP: PT = √(41² − 9²) = √1600 = 40.
3. The alternate segment theorem relates the angle between a tangent and a chord to:
The tangent-chord angle always equals the inscribed angle standing on the opposite side of the chord.
4. Two tangents are drawn to a circle from the same external point. What must be true of their lengths?
The two right triangles formed (each with a radius leg and shared hypotenuse OP) are congruent, so the tangent segments are always equal.
5. The angle between a tangent and a chord at the point of contact is 65°. What is the inscribed angle in the alternate segment?
By the alternate segment theorem, the tangent-chord angle equals the inscribed angle in the alternate segment exactly.
✅ Key takeaways
- A tangent touches a circle at exactly one point and is always perpendicular to the radius there.
- That right angle lets you use the Pythagorean theorem to find tangent lengths from an external point.
- The alternate segment theorem: the tangent-chord angle equals the inscribed angle in the alternate segment.
- Match the tangent-chord angle to the segment on the opposite side of the chord, not the same side.
- Two tangents drawn from the same external point are always equal in length.
🎓 Go deeper: external courses & trusted references
Trusted study sites (US & UK)
Practice & reference
External sites are listed for reference only. This course is independent and has no affiliation with, or endorsement from, the institutions named.