Circles • Topic 2 of 3

Properties of Tangents & Perpendicularity Theorems

What is the Tangent Perpendicularity Theorem? The most vital property of a tangent line involves its angular relationship with the radius. The Tangent Perpendicularity Theorem states: The tangent line at any given point on a circle is always perfectly perpendicular (at a right angle of 90 degrees) to the radius that passes directly through that point of contact.

To understand why this is geometrically true, think about walking along a straight line path next to a circular fence. The single point where you are closest to the center of the circular area is the exact point of contact. In geometry, the shortest path from a fixed point to a straight line is always a straight perpendicular line. Therefore, the radius meeting the tangent line must form a clean 90-degree angle.

The Converse Theorem The reverse of this rule is also entirely true and serves as the Converse of the Tangent Theorem: If a straight line is drawn through the end point of a radius on the circle's boundary such that it is perpendicular to that radius, then that line must be a tangent to the circle.

Number of Tangents from Varying Points The total number of tangent lines you can possibly draw to a circle depends entirely on where you place your pencil tip relative to the circle's boundary:

1. From a point inside the circle: You can draw zero tangents. Any line drawn through an interior point will always cut the circle twice, turning it into a secant. 2. From a point exactly on the circle: You can draw exactly one unique tangent line. 3. From a point outside the circle: You can draw exactly two distinct tangent lines to the circle.

---

DIAGRAM 1: PERPENDICULAR RADIUS-TANGENT RELATIONSHIP

                . - ~ ~ ~ - .
            .                   .
          /                       \
         /            O            \
        ;             |             ;
         \            | Radius     /
          \           |           /
            .         | 90°     .
                . - - X - - .  <-- Point of Contact
                     / \
                    /   \ Tangent Line
                   /_____\

DIAGRAM 2: TANGENT COUNT BASED ON POINT POSITION

   Point Inside (0 Tangents)     Point On (1 Tangent)     Point Outside (2 Tangents)
          .-~~~-.                       .-~~~-.                    .-~~~-.
        .         .                   .         .                .   /     .  \
       /     P     \                 /     P     \              /   /       \  \
      ;             ;               ;======X======;            ;   /         \  ;
       \           /                 \           /              \ /           \ /
        .         .                   .         .                .             .
          '-___-'                       '-___-'                 /               \
                                                               /                 \
                                                              P'                  P''
Solved Examples
4
Worked Example
A radius of length 7 cm is drawn inside a circle centered at point O. A tangent line passes through point P on the circle's edge. A point Q is located along this tangent line such that the straight line distance from O to Q measures 25 cm. Find the exact length of the tangent segment PQ.
Solution
  1. Step 1: Identify the geometric shape formed by the lines.*
  2. According to the Perpendicularity Theorem, the radius OP is perpendicular to the tangent line PQ at the point of contact P.*
  3. Therefore, triangle OPQ is a right-angled triangle with the right angle located at vertex P.*
  4. Step 2: Identify the hypotenuse and legs of the right triangle.*
  5. The side opposite the 90-degree angle is OQ, making it the hypotenuse.*
  6. OP (radius) = 7 cm, OQ (hypotenuse) = 25 cm, and PQ (tangent segment) is the remaining leg.*
  7. Step 3: Apply the Pythagoras theorem to solve for PQ.*
  8. Base squared + Height squared = Hypotenuse squared*
  9. OP squared + PQ squared = OQ squared*
  10. 7 squared + PQ squared = 25 squared*
  11. 49 + PQ squared = 625*
  12. PQ squared = 625 - 49 = 576*
  13. PQ = Square root of 576 = 24 cm.*
  14. Answer: The length of PQ is 24 cm.
5
Worked Example
Point P lies completely outside a circle with a radius of 9 cm. Two lines are drawn from point P. One line is a secant passing through the interior, and the second line is a tangent that skims the circle at point T. If the center of the circle is O, and the distance from P to the closest point on the circle's boundary is 6 cm, calculate the straight line distance from P to the point of contact T.
Solution
  1. Step 1: Find the total length of the line segment connecting center O to external point P.*
  2. The distance from P to the circle boundary is given as 6 cm. The distance from the boundary to the center O is the radius, which is 9 cm.*
  3. Therefore, total distance OP = 9 cm + 6 cm = 15 cm.*
  4. Step 2: Apply the tangent-radius theorem.*
  5. The radius OT is perpendicular to the tangent PT at point T, creating right-angled triangle OTP.*
  6. Step 3: Use the Pythagoras theorem to find PT.*
  7. OT squared + PT squared = OP squared*
  8. 9 squared + PT squared = 15 squared*
  9. 81 + PT squared = 225*
  10. PT squared = 225 - 81 = 144*
  11. PT = Square root of 144 = 12 cm.*
  12. Answer: The distance from P to T is 12 cm.
6
Worked Example
A straight horizontal line XY touches a small circular metallic ring centered at O at exactly one point T. A student measures the angle formed between a chord TA and the tangent line segment TX to be 40 degrees. Calculate the measure of the angle OTA.
Solution
  1. Step 1: Apply the perpendicularity theorem to find the full angle at the point of contact.*
  2. The radius OT is strictly perpendicular to the tangent line XY at point T. Therefore, angle OTX = 90 degrees.*
  3. Step 2: Calculate the angle OTA using adjacent angle subtraction.*
  4. The angle OTX is composed of two adjacent parts: angle OTA and angle ATX.*
  5. We are given that angle ATX = 40 degrees.*
  6. Therefore, angle OTA = angle OTX - angle ATX = 90 degrees - 40 degrees = 50 degrees.*
  7. Answer: Angle OTA is 50 degrees.
  8. --

Key Points

  • The radius connecting to a tangent line at its point of contact always forms a 90-degree perpendicular angle.
  • This perpendicular relationship allows you to solve for unknown lengths using the Pythagoras Theorem.
  • Zero tangents can ever be drawn originating from an interior point located inside a circle.
  • Exactly one tangent line can be constructed passing through a point resting on the boundary.
  • Exactly two tangents can be projected onto a circle from any single exterior point.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The tangent at any point of a circle is ___ to the radius through the point of contact.
Explanation: Tangent $\perp$ radius.
Q2.Tangents drawn from an external point to a circle are:
Explanation: $PA=PB$.
Q3.The angle between a tangent and the radius at the point of contact is:
Explanation: $90^\circ$.
Q4.If $PA$ and $PB$ are tangents from $P$, then:
Explanation: Equal tangent lengths.
Practice more MCQs → 📝 Assignment · 35 marks