Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Geometry → Circles ↓


Number of Tangents from a Point


In the world of geometry, circles are one of the most beautiful and fascinating shapes. Circles are defined as a collection of points that are equidistant from a fixed point called the center. The tangent to a circle is a straight line that touches the circle at exactly one point. This point is known as the tangent point. Tangents are important to understand because they appear in a variety of geometric problems and they have properties that can help solve complex calculations.

Introduction to tangents

Before getting into the main topic of the number of tangents from a point, it is necessary to understand what a tangent actually is. A tangent is a line that intersects a circle at only one point. This feature distinguishes it from a secant line, which intersects a circle at two points.

Mathematically, if you have a circle with center (O) and a tangent line touching the circle at point (A), then (OA) is perpendicular to the tangent line at point (A). This relationship gives rise to some important theorems in circle geometry.

Tangent and radius relation

One of the most basic principles associated with tangents and circles is that the tangent is always perpendicular to the radius at the point of tangency. This can be written as:

If the line ( PT ) is tangent to a circle at point ( T ), then ( OT perp PT ).
Hey Tea

Types of points relative to a circle

When discussing tangents, it is necessary to classify the types of points belonging to the circle:

  1. Inside the circle: When a point lies inside the circle, then it is impossible to draw a tangent to the circle from that point.
  2. On a circle: When a point lies on a circle, there is only one tangent to it, which is the circle itself.
  3. Outside the circle: If a point is outside the circle, then you can draw exactly two tangents from it to the circle. These tangents are equal in length.

Number of tangents from a point outside the circle

The main focus of this topic is that when a point lies outside a circle, two tangents can be drawn. To understand this, consider a circle with center (O) and a point (P) outside the circle.

Hey P B A

From the point (P), tangents (PA) and (PB) can be drawn such that both touch the circle at points (A) and (B) respectively. The important thing is that these points (A) and (B) are where the circle is tangent, and by the definition of tangents, the angles they make with the radii (OA) and (OB) are 90 degrees.

Properties of tangents to a circle from a point

Following are some of the essential qualities:

  1. The length of tangents drawn from an external point to a circle is equal. Hence, (PA = PB).
  2. The tangent at any point of a circle is perpendicular to the radius through the point of contact.
  3. If two tangents are drawn to a circle from an external point, then: 
                ∠OPA = ∠OPB and ∠OAP = ∠OBP = 90°

Understanding more through examples

Example 1: Finding the length of a tangent

Suppose the radius of a circle with center (O) is 5 units. A point (P) outside the circle is 13 units away from (O). Calculate the length of each tangent from (P) to the circle.

To solve this, use the Pythagorean theorem on a right-angled triangle ( triangle OAP ) where ( OA = 5 ), ( OP = 13 ) and ( PA ) is the tangent:

OP² = OA² + PA²
13² = 5² + PA²
169 = 25 + PA²
PA² = 144
PA = √144 = 12

Hence, the length of the tangents (PA) and (PB) is 12 units each.

Example 2: Understanding tangents

A circle has its centre at (O) and a point (P) lies outside the circle such that the distances are (OA = 10) and (OP = 15). If PA and PB are both tangents, how will you classify the points A and B? Verify the length of the tangents.

On the other hand, use the Pythagorean theorem again:

OP² = OA² + PA²
15² = 10² + PA²
225 = 100 + PA²
PA² = 125
PA = √125

Therefore, (PA approx 11.18 ) units, which confirms that (PA = PB) and (A) and (B) are tangent points to the circle.

Real life applications of tangents

Tangents are used not only in theoretical mathematics but also in real-world situations:

  • Navigation: Ships and aircraft use tangents in navigation systems to stay on their course.
  • Architecture: Tangential lines are used in the design of domes and arches.
  • Mechanical engineering: Tangent principles are often used in gear couplings and wheel alignment.

Conclusion

Tangents to a circle are important in understanding circular geometry and solving geometric problems. From a single external point, exactly two tangents can be drawn to a circle, and the lengths of these tangents are equal. This property is invaluable in both theoretical explorations and practical applications, making it an essential concept in mathematics. Understanding tangents deeply enhances problem-solving skills and provides insight into the beauty and applications of geometry.


Grade 10 → 5.3.3


U
username
0%
completed in Grade 10

← Prev (5.3.2)
Tangent to a Circle
Next (5.3.4) →
Angle Subtended by an Arc

Comments 



Number of Tangents from a Point

https://www.buddymath.com/g10/5.3.3?bmal=en