Two tangents, one length
The two tangents from an outside point are equal in length.
Measure both
From an outside point P, two tangents touch the circle at T₁ and T₂. The lengths PT₁ and PT₂ can be measured.
Move P around and compare the two lengths.
How do PT₁ and PT₂ compare?
What this lesson covers
The idea
Theorem 10.2: the lengths of the tangents drawn from an external point to a circle are equal, since the right triangles formed with the radii are congruent (RHS).
Measure both
From an outside point P, two tangents touch the circle at T₁ and T₂. The lengths PT₁ and PT₂ can be measured.
Move P around and compare the two lengths.
How do PT₁ and PT₂ compare?
- They are always equal
- The upper one is longer
- They are equal only for some positions of P
Measure and fold
Drag P. Both tangent lengths are measured. Then press Fold along OP and watch the two triangles.
The tangents are equal
Theorem 10.2. The lengths of the tangents drawn from an external point to a circle are equal.
Join OP, OT₁ and OT₂. The angles at T₁ and T₂ are right angles (Theorem 10.1). OT₁ = OT₂ (radii) and OP is common, so the right triangles OT₁P and OT₂P are congruent by RHS. Hence PT₁ = PT₂. Pythagoras gives the same: PT₁² = OP² − OT₁² = OP² − OT₂² = PT₂².
Notes
Theorem 10.2. The lengths of the tangents drawn from an external point to a circle are equal. The right triangles formed with the radii are congruent (RHS).
Check yourself
Two tangents PQ and PR are drawn to a circle from an external point P. Which is true?
To prove PQ = PR, we show that the right triangles OQP and ORP are congruent. By which rule?
The tangents PA and PB from an external point P have lengths PA = 2x + 3 and PB = 5x − 6 (in cm). Find x.
Answer: 3
PA = PB, so 2x + 3 = 5x − 6, which gives 9 = 3x and x = 3. Check: PA = 9 and PB = 9.
TP and TQ are tangents from T to a circle, and angle PTQ = 80°. Find angle TPQ (in degrees).
Answer: 50
TP = TQ, so angle TPQ = angle TQP = (180° − 80°) ÷ 2 = 50°.
- PQ is longer than PR. They are equal for every position of P.
- PQ = PR — correct. Yes! The lengths of the tangents from an external point are equal (Theorem 10.2).
- PQ + PR = OP. No. PQ and PR are two sides of different triangles. They are not parts of OP.
- SSS, because all three sides are given. We do not know PQ and PR yet. That is what we want to prove.
- RHS: a right angle, the hypotenuse OP, and OQ = OR — correct. Yes! Right angle at Q and R, common hypotenuse OP, and equal radii OQ = OR.
- AAA. Equal angles give similar triangles, not necessarily congruent ones.