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 in one single point never entering the circle's interior is Tangent. A circle can have many tangents. Tangents are perpendicular to the radius. Let us read more about a tangent to circle theorems which plays a significant role in geometrical constructions and proofs.
|1.||What is Tangent?|
|2.||What is Tangent to Circle?|
|4.||Tangent to Circle Theorems|
|5.||FAQs on Tangent|
What is Tangent?
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 in geometry is defined as a line or plane that touches a curve or a curved surface at exactly one point.
What is Tangent to Circle?
A tangent to circle in geometry 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 from 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.
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 a right angle.
Apart from the above-listed properties, a tangent to the circle has mathematical theorems associated with it and are used while doing major calculations in geometry. Let us discuss a few tangents to circle theorems in detail.
Tangent to Circle Theorems
Tangent Theorem I: 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.
Tangent Theorem II: Suppose that two tangents are drawn to a circle S from an exterior point C. Let the points of contact be A and B, as shown:
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.
- OA = OB (radii of the same circle)
- OC = OC (common)
- Thus, by the RHS criterion, ΔCAO is congruent to ΔCBO, and the truth of all the three assertions follows.
☛Topics Related to Tangent
Check out some interesting articles related to the tangent and tangent to the circle.
Tangent to Circle Examples
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?
Given: TP and TQ are tangents.
Using theorem I, if the radius is drawn to the tangents TP and TQ it will be perpendicular to these tangents.
Thus OP perpendicular to TP and QO perpendicular to TQ
∠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°
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, that is, of CA and CB?
Solution: Join OC, and let it intersect AB at D:
Note that ∠ADC=90°. On comparing ΔOAC with ΔODA:
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 cm. We need to calculate now the value of OD using the Pythagoras Theorem:
OD2 = OA2 − AD2 =62 − (10/2)2 =11
OD=√(11) = (3.316)cm
We plug this value into the similarity relation OD:OA = AD:AC to get:
AC=(OA×AD)/OD=(6×10/2)/(3.316) ≈ 9.047cm
This is the (approximate) length of the two tangents CA and CB.
Example 3: Consider two concentric circles of radii 5 cm and 7 cm. 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 must be the midpoint of AB (why?), so AC=BC=(1/2)AB. Using the Pythagoras Theorem, we have:
AC2 = OA2 − OC2 = 72 − 52 = 24
AC = 4.898 cm
AB = 2×AC = 9.796cm
Therefore the length of the tangent AB is 9.796 cm.
FAQs on Tangent
What is the Definition 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.
How Do You Define Tangent to Circle?
A tangent is a line that never enters the circle’s interior. Tangent to circle can be described as straight line beginning from a point on a circle and is perpendicular to the radius. A tangent of the circle touches the circle at one point but not enters 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.
Write General Equations of Tangent to Circle?
General equations of 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 (\(a_1, b_1)\) 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.
What are the Four Properties of Tangents to a Circle?
The four major properties of tangent to circle are listed as follows:
- The tangent is a straight line that touches the circle at one point.
- It is perpendicular to the radius, at the point of tangency.
- A tangent never enters the circle's interior
- The tangent segments to a circle are equal, from the same external point.
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 tangent to circle is must. A tangent can be used for different applications such as:
- In the differentials and approximations
- Strength of materials
How to Construct a Tangent?
To construct tangents on a circle following steps are necessary:
- Step I: Make a line that joins the point to the center of the circle.
- Step II: Draw the perpendicular bisector for the respective line drawn in step I.
- Step III: Put the compass at the center of the circle. Open and adjust its length up to the end-point.
- Step IV: Make an arc through the circle.
- Step V; The point where the arc crosses the circle is the point of tangency or a tangent point.
How Do We Know if Two Circles are Tangent?
We know that a line is considered as a tangent to circle, if the line is touching 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.