What Is a Tangent Line to a Circle Formula and Properties
The tangent of a circle is known as a line touching circles or an ellipse at only one point. Imagine, when a line touches the curve at P, then this point “P” is known as the point of tangency. In differential geometry, the tangent equation can be found using the following methods:
So, we know that finding the gradient of the curve is the gradient of the tangent to the curve at any specified point given on the curve. Hence, the tangent equation of the curve y = f(x) is:
to find the derivative of gradient function through the rules of differentiation.
To find the gradient of the tangent, replace the x- coordinate of the given point in the derivative given.
In the straight-line equation’s slope -point formula, replace the gradient of the tangent and given coordinate point to find out the tangent equation.
Tangent of a Circle Definition
A circle is also known as a curve. It is also a closed two-dimensional shape. It is to be observed that the radius of the circle or the line joining the centre O to the point of tangency or the radius of the circle and tangent line are always perpendicular to each other, i.e. OP is perpendicular to XY as shown in the below figure.
[Image will be Uploaded Soon]
Here “XY” is the tangent of a circle given, and “OP” is the point of tangency and the tangent radius and the point “O” represents the centre of the circle.
Thus, the radius and the tangent to a circle are related to each other, tangent to a circle formula that can be well explained using the tangent theorem.
Tangent Meaning in Trigonometry
The tangent of an angle is called the ratio of the length of measure of the opposite side to the length of the adjacent side's measure. Hence, it is regarded as the ratio of sine and cosine function of an acute angle; however, the value of cosine function should not equal to zero. It is regarded as one of the six primary functions in trigonometry.
[Image will be Uploaded Soon]
The direct common tangent formula is:
Tan P = Opposite Side/Adjacent side
Tangent may be given by using Sine and Cosine as:
Tan P = Sin P / Cos P
The sine of an angle is the length of the measure of the opposite side divided by the length of the hypotenuse side's measure. The cosine of the angle is given by the ratio of the length of the measure of the adjacent side to the ratio of the length of the measure of the hypotenuse side.
So, That is, Sin P = Opposite Side/ Hypotenuse Side
Cos P = Adjacent Side/ Hypotenuse Side
tan P = Opposite Side/ Adjacent Side
In trigonometry, the tangent function will help find the slope of a line between the point representing the intersection, the hypotenuse and the altitude of a right triangle and the origin.
Hence, tangent signifies the slope of some object in Trigonometry and tangent geometry. Now let us see at the most important tangent angle – 30 degrees and its derivation.
Derivation of Value of Tangent 30 Degrees
As per the properties of a right-angle triangle when its acute angle equals 30⁰, then the length of the hypotenuse is double the length of the opposite side. The length of the adjacent side is 3√2 times to the length of the measure of the hypotenuse side.
Hence,
Length of Hypotenuse = 2×Length of the measure of the opposite side
Length of Adjacent side= √3/2 × Length of Hypotenuse
Length of Adjacent side= √3/2 × (2×Length of Opposite side)
Length of Adjacent side= (√3/2×2) ×Length of Opposite side
Length of Adjacent side=√3 × Length of Opposite side
1√3=Length of opposite side/length of the adjacent side
Since the ratio is tan30⁰,
tan30⁰ = 1/√3
Similarly, we can find the values of other angles like 45, 60 using this property of right-angled triangles.
Applications of Tangents in Science and Technology
Tangent has a wide range of use in science and technology as it is the function of sine and cosine. Some of the areas that use trigonometric functions are the Artificial Neural Networks, visualisations, behaviour of elementary particles, and waves like sound waves, electromagnetic waves.
FAQs on Tangents in Geometry Explained with Definitions and Applications
1. What is a tangent in geometry?
A tangent in geometry is a line that touches a curve or circle at exactly one point without crossing it. For a circle, the tangent meets the circle at a single point called the point of contact. At that point, the tangent line is perpendicular to the radius drawn to the point of contact. Tangents are commonly studied in circle geometry and coordinate geometry.
2. What is the formula of a tangent to a circle?
The equation of a tangent to a circle depends on the form of the circle’s equation and the point of contact. For a circle x² + y² = r², the tangent at point (x₁, y₁) on the circle is xx₁ + yy₁ = r². For a general circle (x − a)² + (y − b)² = r², the tangent at (x₁, y₁) is:
- (x₁ − a)(x − a) + (y₁ − b)(y − b) = r²
3. Why is the radius perpendicular to the tangent?
The radius is perpendicular to the tangent at the point of contact because the shortest distance from the center to a line is along the perpendicular. In a circle:
- The radius joins the center to the point of contact.
- The tangent touches the circle at only one point.
- This forms a 90° angle between the radius and the tangent.
4. How do you find the equation of a tangent at a given point?
To find the equation of a tangent at a given point, substitute the point into the standard tangent formula of the circle. Steps:
- Step 1: Verify the point lies on the circle.
- Step 2: Use the formula xx₁ + yy₁ = r² for x² + y² = r².
- Step 3: Simplify the equation.
- 3x + 4y = 25
5. What is the length of a tangent from an external point?
The length of a tangent from an external point to a circle is given by √(d² − r²), where d is the distance from the center to the external point and r is the radius. Formula:
- Tangent length = √(OP² − r²)
- Tangent length = √(100 − 36) = √64 = 8 units
6. What are the properties of tangents to a circle?
The main properties of tangents to a circle describe how they relate to the radius and external points. Key properties include:
- A tangent touches the circle at exactly one point.
- The radius at the point of contact is perpendicular (90°) to the tangent.
- Tangents drawn from the same external point are equal in length.
7. What is the difference between a tangent and a secant?
A tangent touches a circle at exactly one point, while a secant cuts the circle at two distinct points. Differences:
- Tangent: 1 point of contact.
- Secant: 2 points of intersection.
- The radius is perpendicular to the tangent at the point of contact.
8. How many tangents can be drawn to a circle from an external point?
Exactly two tangents can be drawn to a circle from an external point. These two tangents:
- Touch the circle at two different points.
- Have equal lengths.
- Form equal angles with the line joining the center and the external point.
9. Can you give an example of a tangent problem with solution?
Yes, a common tangent problem involves finding the tangent length from an external point. Example:
- Circle radius r = 5 units.
- Distance from center to external point OP = 13 units.
- = √(169 − 25)
- = √144
- = 12 units
10. What is the condition for a line to be tangent to a circle?
A line is tangent to a circle if the distance from the center to the line equals the radius of the circle. Condition:
- If distance from center to line = r → the line is a tangent.
- If distance < r → the line is a secant.
- If distance > r → the line does not intersect the circle.
