Tangent
The word "tangent" means "to touch". The Latin word for the same is "tangere". In general, we can say that the line that intersects the circle exactly at one point on its circumference and never enters the circle's interior is a tangent. A circle can have many tangents. They are perpendicular to the radius. Let us learn more about the tangent meaning and theorems in this article.
| 1. | Tangent Meaning |
| 2. | Tangent of a Circle |
| 3. | Tangent Properties |
| 4. | Tangent Theorems |
| 5. | Tangent of Circle Formula |
| 6. | FAQs on Tangent |
Tangent Meaning
In geometry, a tangent is the line drawn from an external point and passes through a point on the curve. One real-life example of a tangent is when you ride a bicycle, every point on the circumference of the wheel makes a tangent with the road. Let us understand the concept of a tangent with an example. The following figure shows an arc S and a point P external to S. A tangent from P has been drawn to S. This is an example of a representation of a tangent.
Tangent Definition: Tangent in geometry is defined as a line that touches a curve or a curved surface at exactly one point.
Tangent of a Circle
A tangent of a circle is defined as a straight line that touches or intersects the circle at only one point. A tangent is a line that never enters the circle’s interior. The following figure shows a circle with a point P. A tangent L passes through P has been drawn. This is an example of a tangent to circle.
Point of Tangency
The point of tangency is defined as the only point of intersection where the straight line touches or intersects the circle. In the above figure, point P represents the point of tangency.
Tangent Properties
The tangent has two important properties:
- A tangent touches a curve at only one point.
- A tangent is a line that never enters the circle’s interior.
- The tangent touches the circle’s radius at the point of tangency at a right angle.
Apart from the above-listed properties, a tangent to the circle has mathematical theorems associated with it and those theorems are used while doing major calculations in geometry. Let us discuss a few tangents to circle theorems in detail.
Tangent Theorems
There are two most important theorems on the tangent of a circle. Those are the tangent to radius theorem, and the two tangents theorem. Let us discuss their statements and proof in detail.
Tangent Radius Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Given: Tangent PL to a circle S (with the center of circle O), and the point of contact is A.
To prove: OA is perpendicular to the tangent PL.
Proof: Point P lies outside the circle. On joining PO we get PO > OA (radius of a circle). This condition will apply to every point on the line PL except point A.
PO > OA proves that OA is the shortest of all the distances of point O to the other points on PL.
Hence proved, OA is perpendicular to PL.
Two Tangents Theorem: Suppose that two tangents are drawn to a circle from an exterior point C. Let the points of contact be A and B, as shown in the image below.
The theorem states the following:
- The lengths of these two tangents will be equal, that is, CA = CB.
- The two tangents will subtend equal angles at the center, that is, ∠COA = ∠COB.
- The angle between the tangents will be bisected by the line joining the exterior point and the center, that is, ∠ACO = ∠BCO.
Proof: All the three parts will be proved if we show that ΔCAO is congruent to ΔCBO. Comparing the two triangles, we see that:
- OA = OB (radii of the same circle)
- OC = OC (common side)
- ∠OAC = ∠OBC = 90° (Tangent drawn to a circle is perpendicular to the radius at the point of tangency)
- Thus, by the RHS criterion, ΔCAO is congruent to ΔCBO, and the truth of all the three assertions follows.
Tangent of Circle Formula
Let us now learn about the equation of the tangent. Tangent is a line and to write the equation of a line we need two things, slope (m) and a point on the line. General equation of the tangent to a circle:
1) The tangent to a circle equation x2 + y2 = a2 for a line y = mx +c is given by the equation y = mx ± a √[1+ m2].
2) The tangent to a circle equation x2+ y2 = a2 at (\(a_1, b_1)\) is x\(a_1\)+y\(b _1\)= a2
Thus, the equation of the tangent can be given as xa1+yb1 = a2, where (\(a_1, b_1)\) are the coordinates from which the tangent is made.
☛ Related Topics
Check these interesting articles related to the tangent and tangent to the circle.
Tangent to Circle Examples
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Example 1: TP and TQ are the two tangents to a circle with center O such that ∠POQ = 130°, then angle ∠PTQ is equal to?
Solution:
Given: TP and TQ are tangents.
Using the tangent radius theorem, if the radius is drawn to the tangents TP and TQ it will be perpendicular to these tangents.
Thus, OP is perpendicular to TP and QO is perpendicular to TQ. This implies,
∠OPT = 90°
∠OQT = 90°
We know that sum of interior angles of a quadrilateral is 360°.
Therefore, ∠POQ + ∠PTQ + ∠OPT + ∠OQT = 360°
∠PTQ = 360° - (130° + 90° + 90°)
∠PTQ = 50°
Therefore, the value of ∠PTQ is 50°.
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Example 2: Consider a chord AB of length 10 cm in a circle of radius 6 cm. Tangents at A and B intersect at C, as shown below:
What are the lengths of these tangents, CA and CB?
Solution:
Note that ∠ADC=90°. On comparing ΔOAC with ΔODA:
- ∠OAC=∠ODA=90°
- ∠DOA=∠COA (common)
Thus, the two triangles ΔOAC and ΔODA are similar by the AA similarity criterion.
Hence, OD: OA = AD: AC.We know that OA = 6 cm and AD is half of AB, which is 10 cm, so AD is 10/2 = 5 cm. Now we need to calculate the value of OD using the Pythagoras theorem:
OD2 = OA2 − AD2 = 62 − (5)2 = 36 - 25 = 11
OD = √(11) = 3.316 cm
We substitute this value into the similarity relation OD: OA = AD: AC to get:
AC = (OA×AD)/OD = (6×5)/(3.316) ≈ 9.047 cm
As per the two tangents theorem, tangents drawn from an external point to a circle measure the same. Thus, AC = CB.
Therefore, AC = BC = 9.047 cm approximately.
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Example 3: Consider two concentric circles of radii 5 inches and 7 inches. A chord AB of the larger circle touches the smaller circle at C. What is the length of AB?
Solution: Consider the following figure:
Note that since AB is the tangent to the smaller circle at C, OC must be perpendicular to AB. Thus, ΔOAC is right-angled at C. Also, C is the midpoint of AB, so AC = BC = (1/2)AB. Using the Pythagoras theorem, we have:
AC2 = OA2 − OC2 = 72 − 52 = 24
AC = 4.9 inches approx
AB = 2 × AC = 9.8 inches
Therefore, the length of the tangent AB is 9.8 inches approximately.
FAQs on Tangent
What is the Meaning of Tangent?
The "tangent" is derived from the Latin word "tangere", which means "to touch". Tangent in geometry is defined as a line or plane that touches a curve or a curved surface at exactly one point on the boundary of the curve.
What is Tangent of a Circle?
A tangent is a line that never enters the circle’s interior. Tangent to circle can be described as a straight line that passes through a point on a circle and is perpendicular to the radius. A tangent of the circle touches the circle at one point but does not enter the circle's interior.
What are the Two Major Theorems of Tangent to Circle?
The two major tangent to circle theorems are listed below:
- The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- The lengths of the two tangents drawn from an external point to a circle are equal.
What is the Formula for Tangent of a Circle?
The general equation for tangent to circle can be expressed as:
- The tangent to a circle equation x2 + y2 = a2 for a line y = mx +c is given by the equation y = mx ± a √[1+ m2].
- The tangent to a circle equation x2+ y2 = a2 at (a1, b1) is xa1 +yb1 = a2.
Thus, the equation of the tangent can be given as xa1 + yb1 = a2, where (a1, b1) are the coordinates from which the tangent is drawn.
What are the Four Properties of Tangents to a Circle?
The four major properties of a tangent to a circle are listed as follows:
- The tangent is a straight line that touches the circle at only one point.
- It is perpendicular to the radius at the point of tangency.
- It never enters the circle's interior.
- The lengths of two tangents to a circle from the same external point are equal.
How Tangent is Important in Real Life?
It is necessary to study tangents because it allows us to find out the slope of a curved function at a specific point. It is easy to find the slope of a line, but to find out the slope in a curved function, a study of the tangent to a circle is a must. A tangent can be used for different applications such as:
- In the differentials and approximations
- Architecture
- Engineering
- Constructions
How do we Know if Two Circles are Tangent?
We know that a line is considered as a tangent to a circle if it touches the circle exactly at a single point. Similarly, one circle can be tangent to the other circle, if the circles are meeting or touching exactly at one point.