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 → Constructions ↓


Tangent to a Circle


In the world of geometry, a tangent is a line that touches a circle at exactly one point. This point is known as the "tangent point". The tangent to a circle is always perpendicular to the radius drawn at the tangent point. This interesting property helps in solving various geometric problems.

Understanding the concept of tangent to a circle

Imagine a flat surface - a piece of paper. Now, place a circular object, such as a coin, on this paper. The paper is the plane, and the edge of the coin represents the circle. If you take a ruler and touch the edge of the coin in such a way that it just touches the edge and doesn't go into or away from it, you are creating a tangent line.

In mathematical language, the tangent line ((L)) to a given circle ((C)) at a point (P) of the circle satisfies the following properties:

  • (L) touches the circle (C) at exactly one point (P), known as the point of contact.
  • The tangent line is perpendicular to the radius drawn from the centre of the circle to the point of contact.
Center P tangent line

In the above picture:

  • The black circle represents the circle (C).
  • The blue line from the center of the circle to the point (P) shows the radius.
  • The point (P) is the touch point where the green line (tangent) meets the circle.
  • The red line represents another tangent to this circle that is perpendicular to the radius.

Mathematical representation

The equation of a circle with centre ((h, k)) and radius (r) in a coordinate plane is given by:

(x - h)^2 + (y - k)^2 = r^2

The line with the equation (y = mx + c) can be tangent to this circle if and only if the set of equations has exactly one solution where the line equation and the circle equation intersect. This happens when the discriminant of the quadratic equation formed is zero.

Properties of tangents

Let's take a closer look at some properties of tangents to a circle:

Perpendicular radius properties

A basic property of a tangent to a circle is that the tangent is perpendicular to the radius at the point of contact.

R Tea tangent line

Here, the black line (RT) is the radius and the red line is the tangent at point (T).

Length of tangent segment

If two tangents are drawn to a circle from an external point, then the length of both the tangents is equal. Let (T_1) and (T_2) be the points of tangency, then:

PT_1 = PT_2

Alternating clause theorem

Alternate section theorem states that the angle between the tangent and the chord at the point of contact is equal to the angle made in the alternate section.

Constructions involving tangents

In this section, we will explore some geometric construction techniques related to tangents to a circle. Using a compass, straight line, and pencil, let's construct a tangent to a circle from a given external point.

Step-by-step construction

  1. Draw a circle: Draw a circle with center (O) and radius (r).
  2. Select an external point: Mark an external point (A) outside the circle.
  3. Find the midpoint: Using a compass, find the midpoint (M) of (OA). This is done by drawing arcs above and below the line (OA) and then connecting their intersection points.
  4. Draw a circle with center (M): Draw another circle with center (M) and radius (MO). This circle will intersect the original circle at two points.
  5. Find the point of tangency: The original circle will intersect this new circle at two points, say (P) and (Q). These are the points of tangency.
  6. Draw tangents: Draw straight lines (AP) and (AQ). These are tangents to the circle from the point (A).
Hey M A P Why N

According to this construction:

  • The red lines are tangents from the point (A) to the circle at points (P) and (Q).
  • The blue dashed circle helps to identify touch points.

Real life examples of tangents

Understanding tangents is not just limited to geometric problems. They also appear in real-life situations. Here are some examples:

  • Bike Wheels: Bicycle wheels touch the ground at exactly one point, forming a tangent.
  • Eye contact lenses: The lens sits on the surface of the eye, which can be thought of as forming a tangent with the curvature of the eye.
  • Satellite orbits: Satellite paths can also describe a tangent as they orbit around planets.

Learning through examples

Let's look at some examples to strengthen our understanding of tangents to a circle.

Example 1

Problem: Find the slope of the tangent to the circle at the point (5, 1) given by the equation ((x - 2)^2 + (y + 3)^2 = 25).

Solution: First, find the derivative of the equation of the circle. The center of the circle is (2, -3) with radius 5.

The equation of the circle is:

(x - 2)^2 + (y + 3)^2 = 25

The slope of the radius at any point ((x, y)) is the slope between (2, -3) and ((x, y)). The slope of the radius at the point (5, 1) is:

m = (1 + 3) / (5 - 2) = 4/3

The slope of the tangent is the negative reciprocal of this slope (because the radius and tangent are perpendicular):

m_tangent = -3/4

Example 2

Problem: Two tangents are drawn from an external point (A) to a circle with center (O). Prove that (angle OAP = angle OAQ), where (P) and (Q) are the points of tangency.

Solution: When two tangents are drawn to a circle from an external point, then their length is equal.

Hence, (AP = AQ).

Since triangles (OAP) and (OAQ) share the line (OA) and (OP = OQ) as they are radii of the circle, the two triangles are congruent according to RHS (Right Hypotenuse) Criterion.

Hence, (angle OAP = angle OAQ) by corresponding parts of congruent triangles.

The above examples illustrate the basic principles of solving geometric problems involving tangents.

Conclusion

Tangents to a circle are one of the fundamental concepts in geometry. Understanding and constructing tangents requires knowledge of circles, lines, and basic trigonometry. Tangents always maintain a perpendicular relationship with the radius, which is an essential property used in problem-solving. Recognizing patterns, applying the principles of perpendicularity, and employing geometric construction techniques aid in better understanding and visualizing the concept of tangents. Whether in mathematics or in real-life applications, tangents play a vital role in wrapping our world with geometry.


Grade 10 → 5.4.2


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Construction of Similar Triangles
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Tangent to a Circle

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