When two tangent segments are drawn from the same external point to a circle, the segments have equal length. This theorem is useful because it lets you find missing lengths without measuring the circle directly. In a diagram with external point P and tangent points A and B, the result is PA = PB.
The equal tangents also create a symmetric kite-shaped quadrilateral when the radii OA and OB are drawn.
Understanding Geometry: Two Tangents from a Point
A tangent touches a circle at one exact location without passing through its interior. That detail matters. A line that cuts through the circle is a secant, so it follows different length rules.
Start with an external point, meaning a point outside the circle. From that point, there are usually two possible tangent paths around the circle.
Their contact points sit on opposite sides of the line joining the external point to the center. This arrangement gives the figure a built in mirror balance, even when the drawing does not look perfectly symmetrical.
The reason for the equal lengths comes from two right triangles hidden in the diagram. Draw segments from the center to each contact point. Each radius meets its tangent at a right angle.
The two triangles have radii of the same length, and they share the segment from the center to the external point. A right triangle is fixed when its hypotenuse and one leg are fixed. Therefore the triangles must match exactly.
Their remaining sides match as well. This is more than a diagram fact. It shows why the result works for every circle, every external point, and every orientation of the picture.
The same right triangles lead to a useful distance relationship. The segment from the center to the external point is the hypotenuse. The radius is one leg.
The tangent segment is the other leg. By the Pythagorean theorem, the tangent length is the square root of the distance from the center to the external point squared minus the radius squared. This relationship only makes sense when the external point is at least one radius away from the center.
If it is closer, it lies inside the circle and no real tangent segment can be drawn. If it is exactly one radius away, the point lies on the circle and there is only one zero length tangent segment.
Students often use this theorem in algebra problems. Expressions may be written for the two tangent lengths, then those expressions can be set equal. The important step is to identify the shared outside point.
Two tangent segments are equal only when they begin at the same external point and touch the same circle. Tangents drawn from different outside points have no reason to match. After solving for a variable, substitute it back into both expressions.
This checks that both lengths are positive and equal. A negative length signals an algebra error or an invalid value.
Angle information can be connected to this picture too. The line from the center to the external point splits the angle between the two tangents into two equal angles. This follows from the matching right triangles.
That angle symmetry is useful in longer proofs involving kites, angle bisectors, or circumscribed polygons. When reading a diagram, do not trust its appearance alone.
Mark right angles at contact points, mark equal radii, and locate the one external point first. Those marks reveal the structure needed for a reliable proof.
Key Facts
- If PA and PB are tangents from the same external point P, then PA = PB.
- A radius to a point of tangency is perpendicular to the tangent line, so OA ⊥ PA and OB ⊥ PB.
- The quadrilateral OAPB is a kite because OA = OB and PA = PB.
- Right triangles OAP and OBP are congruent by HL because OP is shared and OA = OB.
- Tangent length formula from external point: PA = sqrt(OP^2 - r^2), where r is the circle radius.
- If PA = 3x + 2 and PB = 5x - 8, then set 3x + 2 = 5x - 8 to solve.
Vocabulary
- Tangent
- A tangent is a line or segment that touches a circle at exactly one point.
- Point of tangency
- The point of tangency is the single point where a tangent touches a circle.
- External point
- An external point is a point located outside a circle.
- Radius
- A radius is a segment from the center of a circle to any point on the circle.
- Kite
- A kite is a quadrilateral with two pairs of adjacent equal sides.
Common Mistakes to Avoid
- Setting the tangent segments unequal, such as PA > PB, is wrong because tangents drawn from the same external point are congruent.
- Forgetting that the radius is perpendicular to the tangent is wrong because the right angle is the key fact that allows right-triangle reasoning.
- Using the diameter instead of the radius in PA = sqrt(OP^2 - r^2) is wrong because the right triangle uses OA or OB as one leg, and that length is the radius.
- Assuming any two segments from P to the circle are equal is wrong because the theorem applies only to tangent segments, not secants or chords.
Practice Questions
- 1 From external point P, tangents PA and PB touch a circle at A and B. If PA = 14 cm, what is PB?
- 2 A circle has radius 6 cm and OP = 10 cm. If PA is tangent from P to the circle, find PA using PA = sqrt(OP^2 - r^2).
- 3 Explain why quadrilateral OAPB forms a kite when PA and PB are tangents from P and O is the center of the circle.