Ninth Grade Mathematics: Introduction to Tangents and Normals
A straight line and a circle can interact in three distinct ways in a plane: the line can miss the circle entirely, it can cut across the circle at two points as a secant line, or it can touch the circle at exactly one single point as a tangent line. In this chapter, you will master the foundational theorems governing tangents, normals, and radius perpendicularity.
Definition of Tangent, Secant, and Point of Contact
Consider a line l and a circle C(O, r) lying in the same plane:
Line-Circle Intersections:
NON-INTERSECTING LINE SECANT LINE TANGENT LINE
Line l Line l Line l
------------- -----+---------+----- -----+-----*-----+-----
/ \ P (Point of Contact)
. - ~ . . - ~ . . - ~ .
. ' ' . . ' ' . . ' ' .
/ * \ / * \ / * \
; O ; ; O ; ; O ;
\ / \ / \ /
' . _ _ . ' ' . _ _ . ' ' . _ _ . '
Zero intersections Two intersections ONE intersection
A secant line intersects the circle at two distinct points. A tangent line touches the circle at exactly one single point. This unique point is called the point of contact (or point of tangency). The word 'tangent' originates from the Latin verb tangere, meaning 'to touch'.
The Tangent-Radius Perpendicularity Theorem
The primary theorem of tangent geometry establishes that the radius drawn to the point of contact is strictly perpendicular to the tangent line:
The Tangent-Radius Theorem:
The tangent at any point of a circle is perpendicular to the radius
through the point of contact.
In Circle C(O, r):
If line AB is tangent to the circle at point P,
then Radius OP is perpendicular to Tangent AB (OP perp AB).
. - ~ ~ - .
. ' ' .
/ \
; * O ;
| | |
\ | r /
\ | /
. _ | _ '
Tangent Line A ----------------------*---------------------- B
P (Point of Contact)
OP perp AB
Deductive Proof of the Theorem
We prove this theorem using the principle of shortest distance:
Given: A circle with center O, and line AB tangent to the circle at point P.
To Prove: OP is perpendicular to AB.
Proof:
Step 1: Take any point Q on line AB other than the point of contact P.
Step 2: Since AB is a tangent line, it touches the circle ONLY at point P.
Therefore, point Q must lie strictly outside the circle.
Step 3: Because Q lies outside the circle, the distance from center O to Q
must be strictly greater than the radius r:
OQ > r ===> OQ > OP (since OP = r).
Step 4: This inequality holds true for EVERY point Q on line AB except point P:
OP is the shortest distance from the center O to the line AB.
Step 5: In Euclidean geometry, the shortest distance from a point to a line
is the perpendicular distance.
Step 6: Therefore, OP must be perpendicular to AB.
The proof is complete. The radius drawn to the point of contact is always normal (perpendicular) to the tangent.
The Normal to a Circle
In geometry, a normal to a curve at a given point is the straight line perpendicular to the tangent line at that point of contact.
Theorem:
The normal to a circle at any point of contact passes directly
through the center of the circle.
Because the radius OP is perpendicular to the tangent at P, the unique perpendicular line erected at P must coincide with the radius line, and therefore must pass directly through the center O.
Number of Tangents from a Point
The location of a reference point P relative to a circle dictates how many tangents can be drawn from that point:
Point Location Number of Tangents That Can Be Drawn
---------------------------------------------------------------------------------
Inside the circle (d < r) ZERO tangents (Every line through P is a secant)
On the circle (d = r) EXACTLY ONE tangent (At point P itself)
Outside the circle (d > r) EXACTLY TWO tangents of equal length
Two Tangents from an External Point P:
A (Point of Contact)
*
/ | \
AP / | \
/ | \
P *-----* * O \
\ | /
BP \ | /
\ | /
*
B (Point of Contact)
AP == BP
When two tangents AP and BP are drawn from an external point P to a circle, their lengths are strictly equal: AP = BP.
Chapter Exercises
Exercise 1: A tangent line PQ is drawn to a circle of radius 6 cm from an external point P. If the distance from the center O to point P is 10 cm, calculate the length of the tangent segment PQ.
Exercise 2: Prove that the tangents drawn at the ends of a diameter of a circle are parallel to each other.
Exercise 3: Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle that touches the smaller circle as a tangent.
Solutions and Explanations
Solution 1: Let Q be the point of contact on the circle. By the Tangent-Radius Theorem, the radius OQ is perpendicular to the tangent line PQ at the point of contact: Angle OQP = 90 degrees. Triangle OQP is therefore a right-angled triangle with hypotenuse OP = 10 cm and leg OQ = r = 6 cm. By the Pythagorean theorem: PQ^2 + OQ^2 = OP^2, so PQ^2 + 6^2 = 10^2, which gives PQ^2 + 36 = 100. Subtract 36: PQ^2 = 64. Taking the square root gives tangent length PQ = 8 cm.
Solution 2: Let AB be a diameter of a circle with center O. Let line l be the tangent to the circle at endpoint A, and let line m be the tangent to the circle at endpoint B. By the Tangent-Radius Theorem, the radius OA is perpendicular to tangent line l, so Angle 1 = 90 degrees. Similarly, radius OB is perpendicular to tangent line m, so Angle 2 = 90 degrees. Since A, O, and B lie on a straight line (the diameter), diameter AB acts as a transversal line intersecting the two tangent lines l and m. The consecutive interior angles on the same side of transversal AB sum to Angle 1 + Angle 2 = 90 + 90 = 180 degrees. By the Converse of the Consecutive Interior Angles Theorem, because the consecutive interior angles are supplementary, line l must be parallel to line m. The tangents at the ends of a diameter are parallel.
Solution 3: Let the concentric circles have common center O and radii R = 5 cm and r = 3 cm. Let chord AB of the larger circle touch the smaller circle at point P. Because AB touches the smaller circle at P, line AB is a tangent to the smaller circle at point P, and OP is the radius of the smaller circle: OP = r = 3 cm. By the Tangent-Radius Theorem, OP is perpendicular to AB (Angle OPB = 90 degrees). In the larger circle, AB is a chord, and OP is perpendicular to chord AB. The perpendicular from the center to a chord bisects the chord, so P is the midpoint of AB: AP = PB. In right triangle OPB, hypotenuse OB is the radius of the larger circle (OB = R = 5 cm) and leg OP = 3 cm. By the Pythagorean theorem: PB = sqrt(OB^2 - OP^2) = sqrt(5^2 - 3^2) = sqrt(25 - 9) = sqrt(16) = 4 cm. Since P bisects AB, the total chord length is AB = 2 * PB = 2 * 4 = 8 cm.