Properties of Tangents & Perpendicularity Theorems

What is the Tangent Perpendicularity Theorem? The most important property of a tangent concerns its angle with the radius. Theorem (Tangent ⊥ Radius): The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact. In symbols, if a tangent touches a circle (centre $O$) at $P$, then $OP\perp$ tangent, i.e. the angle between them is $90^{\circ}$.

Proof. Let the tangent touch the circle at $P$, and let $O$ be the centre. Take any other point $Q$ on the tangent (other than $P$). Since the tangent meets the circle only at $P$, every point of the tangent except $P$ lies outside the circle. Hence $Q$ is outside the circle, so $OQ>OP$ (a point inside the circle is closer to the centre than the radius; a point outside is farther). This is true for every choice of $Q$ on the line. Therefore $OP$ is the shortest distance from $O$ to the tangent. But the shortest distance from a point to a line is the perpendicular distance. Hence $OP\perp$ tangent, which proves $\angle OPQ = 90^{\circ}$. Q.E.D.

The Converse Theorem. The reverse is also true: if a line through a point $P$ on the circle is perpendicular to the radius $OP$, then that line is a tangent to the circle. Reason: since $OP$ is the shortest distance to the line, every other point of the line is farther than $r$ from $O$, hence outside the circle — so the line meets the circle only at $P$, which is the definition of a tangent.

Tangent length from an external point. Because $OP\perp PT$ (where $PT$ is a tangent touching at $T$ and $P$ is external), triangle $OTP$ is right-angled at $T$. By Pythagoras, $OP^2 = OT^2 + PT^2$, so the tangent length is

$PT = \sqrt{OP^2 - OT^2} = \sqrt{d^2 - r^2}$,

where $d=OP$ is the distance of the external point from the centre and $r$ is the radius. This single formula powers most numerical questions in this chapter.

Number of tangents from varying points. (1) From a point inside the circle: zero tangents, since every line through an interior point cuts the circle twice. (2) From a point on the circle: exactly one tangent (perpendicular to the radius at that point). (3) From a point outside the circle: exactly two tangents.

A useful chord result (two concentric circles). If a chord of a larger circle touches a smaller concentric circle, the point of contact is the midpoint of that chord, and the common-centre radius to the contact point is perpendicular to the chord. This lets you split the chord into two equal halves and apply Pythagoras — a favourite board exam set-up.

Why “perpendicular” matters numerically. Every standard distance computation in this chapter is just Pythagoras applied to the right angle at the point of contact. Memorise the three rearrangements of $OP^2 = r^2 + PT^2$: the tangent length $PT=\sqrt{OP^2-r^2}$; the radius $r=\sqrt{OP^2-PT^2}$; and the centre distance $OP=\sqrt{r^2+PT^2}$. Knowing which quantity is the hypotenuse ($OP$, always opposite the right angle at the contact point) prevents the single most common arithmetic error in board answers.

Tangents and symmetry. The converse theorem is what justifies the standard ruler-and-compass construction of a tangent at a given point: draw the radius to that point, then erect a perpendicular there. Because perpendicular-to-radius forces tangency, the construction is guaranteed correct without any further measurement — a clean example of a theorem doing practical work.

Common mistakes to avoid. (i) The perpendicular holds only at the point of contact, not at any random point of the tangent. (ii) When applying Pythagoras, $OP$ (centre to external point) is always the hypotenuse; mixing it up with the tangent length gives a wrong, larger answer. (iii) The distance “from the point to the circle” is not the same as $OP$; you must add the radius: $OP=(\text{distance to circle})+r$. (iv) Always confirm the right angle is at the point of contact before writing $a^2+b^2=c^2$.