Circles
Tangent to a Circle
What is a circle? A circle is a collection of all points in a flat plane that are at a constant, fixed distance from a central fixed point. Think of a giant Ferris wheel spinning around its central axle, a round dinner plate, or the boundary of a circular coin. The path traced by these boundaries represents a perfect mathematical circle.
Let us review the foundational parts of a circle that you must remember:
- Centre: The fixed point in the exact middle of the circle.
- Radius: The straight line segment connecting the centre to any point on the boundary.
- Diameter: A straight line passing straight through the centre, connecting two opposite points on the boundary. It is exactly twice the length of the radius.
- Chord: A straight line segment connecting any two points on the circle's boundary. The diameter is simply the longest possible chord!
- Arc: A continuous portion or segment of the circular boundary line.
- Circumference: The total perimeter or outer boundary distance around the entire circle.
Regions of a Circle A circle divides a flat plane into three distinct parts:
1. The interior: The space safely trapped inside the circular boundary (like the cheese on top of a pizza). 2. The exterior: The entire infinite space outside the boundary line. 3. The circle itself: The exact boundary line where points lie at an exact radius distance from the centre.
Lines Intersecting a Circle When a straight flat line crosses paths with a circle, only three scenarios can occur:
- Non-intersecting Line: The line passes completely outside the circle without touching it.
- Secant: A straight line that cuts directly through a circle, intersecting it at exactly two distinct points. It behaves like an arrow piercing a target.
- Tangent: A highly special straight line that just brushes against the outer edge of the circle, touching it at exactly one single, unique point.
The single spot where a tangent line meets the circle is called the point of contact.
| Line Type | Number of Shared Points | Relationship with Circle |
|---|---|---|
| Non-intersecting | 0 | Passes completely outside the boundary |
| Secant | 2 | Cuts through the interior, forming a chord |
| Tangent | 1 | Glides past the exterior, skimming one point |
---
DIAGRAM 1: FOUNDATIONAL ANATOMY OF A CIRCLE
. - - ~ ~ ~ - - .
. / .
/ / \
/ / Radius \
; O--------------------;
; / Centre ;
\ / /
\ / Arc /
. _ _ _ _ _ _ _ _ _ _ .
Chord
DIAGRAM 2: SECANT VS TANGENT LINE PATHS
Secant Line (2 Points)
\ /
\ / . - ~ ~ ~ - .
\ / . .
X / \
/ \ ; ;
/ \ ; ;
/ \ \ /
/ \ . . <-- Point of Contact
\ ' - _ _ _ _X
\ /
\ / Tangent Line (1 Point)
DIAGRAM 3: REGIONS OF A CIRCLE
[ Exterior Space ]
. - - ~ ~ ~ - - .
. .
/ \
/ [ Interior Space ] \
; O ;
\ /
\ /
. .
. - - _ _ _ _ - - .
- Step 1: Analyze the geometric relationship given in the problem statement.*
- The perpendicular distance from the central point of the circle to the passing line is exactly equal to the circle's radius length.*
- Step 2: Apply definitions of intersecting lines.*
- If the distance were less than the radius, the line would penetrate the interior (secant).*
- If the distance were greater than the radius, the line would miss the circle completely (non-intersecting).*
- Since the distance matches the radius perfectly, the line touches exactly one point on the outermost boundary edge.*
- Step 3: Conclude the line type.*
- A line intersecting a circle at exactly one single point is structurally defined as a tangent.*
- Answer: The line is a tangent.
- Step 1: Calculate the radius of the circle from the given diameter.*
- Radius = Diameter / 2 = 16 cm / 2 = 8 cm.*
- Step 2: Compare the center-to-line distance with the calculated radius.*
- The measured distance from the center to the line is 6 cm.*
- Since 6 cm is strictly less than the radius of 8 cm, the line passes inside the boundary of the circle.*
- Step 3: Determine the intersection points and classification.*
- A straight line entering the interior of a circle must cross the boundary line at exactly two distinct points.*
- A line that cuts through a circle at two points is classified as a secant.*
- Answer: 2 points; it is a secant.
- Step 1: Deduce the radius of the clock from the geometric properties.*
- The wooden strip shares exactly one point with the circular rim, meaning it acts as a geometric tangent.*
- The shortest distance from a point to a line is always the perpendicular distance.*
- For a tangent line, this shortest distance to the center matches the radius of the circle exactly.*
- Therefore, Radius (r) = 15 cm.*
- Step 2: Use the standard circumference formula.*
- Circumference = 2 (22 / 7) r*
- Step 3: Substitute the radius value and evaluate.*
- Circumference = 2 (22 / 7) 15*
- Circumference = 44 15 / 7
- Circumference = 660 / 7 cm = 94.28 cm.*
- Answer: 660/7 cm (or approximately 94.28 cm).
- --
- A circle is a 2D path of points maintained at an identical radius distance from a single central point.
- A chord links any two boundary points, while a diameter is a central chord measuring double the radius.
- A secant cuts straight through a circle, intersecting its perimeter at exactly two distinct points.
- A tangent skims past the outside of the circle, sharing exactly one singular point called the point of contact.
- Real-world wheels, gears, and horizons are practical everyday examples of circles and tangent lines interacting.
Number of Tangents
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''
- Step 1: Identify the geometric shape formed by the lines.*
- According to the Perpendicularity Theorem, the radius OP is perpendicular to the tangent line PQ at the point of contact P.*
- Therefore, triangle OPQ is a right-angled triangle with the right angle located at vertex P.*
- Step 2: Identify the hypotenuse and legs of the right triangle.*
- The side opposite the 90-degree angle is OQ, making it the hypotenuse.*
- OP (radius) = 7 cm, OQ (hypotenuse) = 25 cm, and PQ (tangent segment) is the remaining leg.*
- Step 3: Apply the Pythagoras theorem to solve for PQ.*
- Base squared + Height squared = Hypotenuse squared*
- OP squared + PQ squared = OQ squared*
- 7 squared + PQ squared = 25 squared*
- 49 + PQ squared = 625*
- PQ squared = 625 - 49 = 576*
- PQ = Square root of 576 = 24 cm.*
- Answer: The length of PQ is 24 cm.
- Step 1: Find the total length of the line segment connecting center O to external point P.*
- 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.*
- Therefore, total distance OP = 9 cm + 6 cm = 15 cm.*
- Step 2: Apply the tangent-radius theorem.*
- The radius OT is perpendicular to the tangent PT at point T, creating right-angled triangle OTP.*
- Step 3: Use the Pythagoras theorem to find PT.*
- OT squared + PT squared = OP squared*
- 9 squared + PT squared = 15 squared*
- 81 + PT squared = 225*
- PT squared = 225 - 81 = 144*
- PT = Square root of 144 = 12 cm.*
- Answer: The distance from P to T is 12 cm.
- Step 1: Apply the perpendicularity theorem to find the full angle at the point of contact.*
- The radius OT is strictly perpendicular to the tangent line XY at point T. Therefore, angle OTX = 90 degrees.*
- Step 2: Calculate the angle OTA using adjacent angle subtraction.*
- The angle OTX is composed of two adjacent parts: angle OTA and angle ATX.*
- We are given that angle ATX = 40 degrees.*
- Therefore, angle OTA = angle OTX - angle ATX = 90 degrees - 40 degrees = 50 degrees.*
- Answer: Angle OTA is 50 degrees.
- --
- 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.
Lengths of Tangents
What is the Length of a Tangent Theorem? When you choose a point completely outside a circle, you can draw exactly two tangents to it. The Lengths of Tangents Theorem states: The lengths of these two tangent segments drawn from an external point to a circle are perfectly equal to each other.
To prove this key property, imagine drawing lines from the external point to the center of the circle, along with the two radii to the points of contact. This creates two distinct right-angled triangles sharing the same hypotenuse. Because both triangles have a right angle, identical radii lengths, and share a common side, they are completely congruent by the RHS (Right angle-Hypotenuse-Side) rule.
As a direct consequence of this triangle congruence:
- The two outer tangent segments are identical in length.
- The line joining the external point to the center perfectly bisects the angle between the two tangents.
- The line joining the external point to the center perfectly bisects the angle between the two radii.
Quadrilateral and Angle Properties When you trace the two radii and the two external tangents together, they form a closed four-sided polygon (a quadrilateral). Since two corners of this quadrilateral are right angles (90 degrees each), the remaining two opposite angles (the angle between the tangents and the angle between the radii) must add up to exactly 180 degrees. This means they are supplementary angles.
---
DIAGRAM 1: CONGRUENT TRIANGLES FORMED BY EXTERNAL TANGENTS
T1 (Point of Contact 1)
/\
/ \ Radius (r)
/ \
/ \
External /________\ O (Center)
Point P \ /
\ /
\ / Radius (r)
\ /
\/
T2 (Point of Contact 2)
Triangle PT1O is perfectly congruent to Triangle PT2O. Therefore, PT1 = PT2.
DIAGRAM 2: SUPPLEMENTARY ANGLES IN TANGENT QUADRILATERALS
T1
X\
/ \ 90°
/ \
Angle /______\ Center O (Angle y)
(x) \ /
\ / 90°
\ /
X/
T2
Angle (x) + Angle (y) + 90° + 90° = 360° => Angle (x) + Angle (y) = 180°
DIAGRAM 3: RULER AND COMPASS CONSTRUCTION CONCEPT
T1
/ \
/ \ [Draw Circle with diameter OP]
/ . \
P . O .
\ . /
\ /
\ /
T2
- Step 1: Recall the fundamental theorem regarding external tangents.*
- The theorem states that the lengths of two separate tangents drawn from the exact same external point to a single circle must be equal.*
- Step 2: Match the segments to the theorem statement.*
- The segments PA and PB are both tangents drawn from the identical external point P.*
- Therefore, length of PB must equal the length of PA.*
- Step 3: State the final dimension.*
- Since PA = 14 cm, it must be true that PB = 14 cm.*
- Answer: The length of PB is 14 cm.
- Step 1: Identify the geometric structure and angle relationships.*
- The lines OA, AP, PB, and BO form a quadrilateral OAPB.*
- The angles OAP and OBP are both 90-degree right angles because radii are perpendicular to tangents.*
- Step 2: Apply the supplementary angle rule for external tangent configurations.*
- The sum of the central angle (Angle AOB) and the external angle between the tangents (Angle APB) is always exactly 180 degrees.*
- Angle AOB + Angle APB = 180 degrees*
- Step 3: Substitute the known value and solve for the unknown angle.*
- Angle AOB + 70 degrees = 180 degrees*
- Angle AOB = 180 degrees - 70 degrees = 110 degrees.*
- Answer: Angle AOB is 110 degrees.
- Step 1: Use the external tangents theorem at each individual vertex of the triangle.*
- Vertex A acts as an external point yielding two equal tangents: AD = AF = 5 cm.*
- Vertex B acts as an external point yielding two equal tangents: BD = BE = 6 cm.*
- Vertex C acts as an external point yielding two equal tangents: CE = CF = 7 cm.*
- Step 2: Determine the complete length of each of the three main sides of the triangle.*
- Side AB = AD + BD = 5 cm + 6 cm = 11 cm*
- Side BC = BE + CE = 6 cm + 7 cm = 13 cm*
- Side CA = AF + CF = 5 cm + 7 cm = 12 cm*
- Step 3: Sum the side lengths to calculate the total perimeter.*
- Perimeter = Side AB + Side BC + Side CA*
- Perimeter = 11 cm + 13 cm + 12 cm = 36 cm.*
- Answer: The perimeter of triangle ABC is 36 cm.
- --
- Tangent segments drawn onto a single circle from the same external point share identical lengths.
- The line connecting the external point to the center acts as a perfect angle bisector for the system.
- The angle between the two tangents and the angle between the corresponding radii are supplementary (sum to 180 degrees).
- Triangles inscribed or circumscribed around circles can be solved easily by splitting sides into pairs of equal tangent segments.
- To construct these tangents using a compass, you find the midpoint of the line to the center and draw an auxiliary circle.