Draw a Circle.through P.and Tangential.to.the Line at Q
We detect numerous circular objects in our environs, such as a circular clock, coins, frisbees, train wheels on the runway, and so on. A Tangent of a Circle is a line that touches the circle's boundary at exactly one point. The tangential signal is the place where the line and the circle run across.
The word "tangent" is derived from the Latin word "tangere" (which means "to touch"), which was coined by a Danish mathematician named 'Thomas Fineko' in the early 1800s (1583). Using examples, this commodity will describe a tangent to a circle and illustrate its qualities.
In this article, we will dive deep into the concept of Tangent of a Circle, Definition, Backdrop, Examples, etc. Continue reading to know more.
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Tangent of a Circle Definition
In our daily life, we discover that ii lines either intersect in a plane or non intersect at all. For instance, two parallel lines in a railroad runway never intersect each other, whereas, in windows grill design, the grills intersect each other. Practice you lot know what happens if a curve and a line are given in a unmarried plane? The curve tin can be a parabola, a circumvolve, or any general curve.
Similarly, what will happen if nosotros consider the intersection of a line and a circle? We may become 3 situations equally given in the post-obit diagrams.
| Figure 1 | Figure 2 | Effigy 3 |
| (i) Straight line \(PQ\) does not touch on the circumvolve. | (i) Straight line \(PQ\) touches the circle at a common point \(A\). | (i) Direct line \(PQ\) intersects the circle at two points, \(A\) and \(B\). |
| (ii) There is no mutual point between the direct line and circumvolve. | (ii) \(PQ\) is called the tangent to the circumvolve at \(A\). | (ii) The line \(PQ\) is called a secant of the circumvolve. |
| (iii) The number of points of intersection of a line and circle is zero. | (3) The number of points of intersection of a line and circle is one. | (three) The number of points of intersection of a line and circle is two. |
Articulate YOUR CONCEPTUAL DOUBTS ON TANGENT OF CIRCLE
Definition: A tangent to a circumvolve is a line that touches the circle at only one point. And the point of contact is known equally the point of tangency.
Hither, \(AB\) is the center of the circle, and \(P\) is the point of tangency.
Real-life Examples of Tangent of a Circle
(i) When a cycle moves along a road, then the road becomes the tangent at each point when the wheels whorl on information technology.
(two) When a stone is tied at one end of a string and rotated from the other end, the stone will follow a circular path. If we suddenly stop the motion, the rock will go in a direction tangential to the round motion.
(iii) Abstruse limerick of xanthous brawl and curb, placed tangentially, with the project of spherical shadow, on a stone background is also an case of a tangent of a circumvolve.
Uses of Tangent of a Circumvolve
Circles and tangent lines can be helpful in many real-world applications and fields of written report, such as construction, landscaping, and engineering.
Number of Tangents From a Signal on a Circle
Draw a circle on paper. Take a bespeak \(P\) inside the circle. Tin can a tangent exist fatigued to the circle through this betoken \(P?\)
We see that all the lines through this signal \(P\) intersect the circle at ii distinct points. And so, it is incommunicable to describe any tangent to a circle from a signal inside the circle.
Now, take a point \(P\) on the circle and describe tangents through this point. We observe that only one tangent can be drawn at any betoken on the circle.
Finally, take a point \(P\) outside the circle and try to draw tangents to the circle from this bespeak. What do we notice? We find that two tangents can be drawn to the circle through this point \(P.\)
The length of the tangent segment from the external indicate \(P\) and the betoken of contact with the circle is called the length of the tangent from the indicate \(P\) to the circumvolve.
\(PA\) and \(PB\) are the lengths of the tangents from \(P\) to the circumvolve.
Practice Exam Questions
Tangent of a Circle Method
Case ane: To draw only 1 tangent line
Consider a circumvolve with a heart \(O\) and draw a line perpendicular to the circumvolve'southward radius from a point on the circle. That perpendicular line is called the tangent to the circumvolve.
Case 2: To describe two tangent lines
Consider a circle with a centre \(O\) and draw two lines perpendicular to the circle's radius from two distinct points on the circle. That perpendicular lines are called the tangent to the circle.
Properties of Tangent of a Circle
Theorem 1 : The tangent at any signal of a circle is perpendicular to the radius through the point of contact.
Given: A circle \(C\left( {0,\,r} \right)\) and a tangent \(I\) at indicate \(A\).
Let us endeavour to prove \(OA \bot I\)(\(OA\) is perpendicular to \(I\))
Construction: Take a betoken \(B,\) other than \(A,\) on the tangent \(I.\) Join \(OB.\) Suppose \(OB\) meets the circumvolve at point \(C\).
\(OA=OC\) (Radius of the same circle)
Now, \(OB = OC + BC.\)
\(∴ OB > OC\)
\(OB > OA\)
\(OA < OB\)
B is an arbitrary point on the tangent \(I\). Thus, \(OA\) is shorter than whatsoever other line segment joining \(O\) to any point on \(I\).
Here, \(OA \bot I\), the converse of the tangent theorem.
Theorem 2 : A line perpendicular to the radius at its signal on the circumvolve is a tangent to the circle.
Given: \(M\) is the centre of a circle, and \(MN\) is the radius.
Line \(l \bot\) seg \(MN\) at \(N\)
To Prove: Line \(l\) is a tangent to the circumvolve.
Proof: Take whatever point \(P\), other than \(Northward\), on the line \(l\). Describe seg \(MP\).
Now in \(\Delta MNP,\angle N\) is a right bending.
Therefore, seg \(MP\) is the hypotenuse.
Therefore, \(segMP > {\mathop{\rm seg}\nolimits} MN\)
As seg \(MN\) is the radius, point P cannot be on the circumvolve.
So, no other point except indicate \(N\), of line \(fifty\) is on the circle.
Line \(l\) intersects the circle in merely one point \(N\).
Therefore, line \(50\) is tangent to the circle.
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Theorem 3 : The lengths of tangents drawn from an external point to a circle are equal.
Given: a circle with heart \(O\), a point \(P\) lying outside the circumvolve and two tangents \(PA, PB\) on the circumvolve from \(P\).
To Prove: \(PA=PB\)
Construction: Join \(OA, OB\) and \(OP\).
Proof: \(\bending OAP = \angle OBP = {xc^ \circ }\) (Because a tangent at whatsoever point of a circle is perpendicular to the radius through the indicate of contact)
At present in right \(\Delta OAP\) and \(\Delta OBP\)
\(OA=OB\) (Radii of the same circle)
\(OP=OP\) (Common)
Therefore, \(\Delta OAP \cong \Delta OBP\) (Past RHS congruency criteria)
Hence, \(PA=Lead\) (By CPCT)
Solved Examples – Tangent of a Circumvolve
Q.ane. In the given figure, a circle with centre \(D\) touches the sides of \(\bending ACB\) at \(A\) and \(B\). If \(\angle ACB = {52^ \circ }\), notice the measure of \(\angle ADB\).
Ans: Nosotros know that the sum of all angles of a quadrilateral is \({360^ \circ }\)
\(\angle ACB + \angle CAD + \angle CBD + \bending ADB = {360^ \circ }\)
\(\Rightarrow {52^ \circ } + {90^ \circ } + {90^ \circ } + \angle ADB = {360^ \circ }\) (By using the tangent theorem)
\( \Rightarrow \angle ADB + {232^ \circ } = {360^ \circ }\)
\(\therefore \angle ADB = {360^ \circ } – {232^ \circ } = {128^ \circ }\)
Q.2. Find the length of the tangent drawn from a point whose distance from the center of a circle is \({\rm{5}}\,{\rm{cm}}\) and radius of the circle is \({\rm{3}}\,{\rm{cm}}{\rm{.}}\)
Ans:
Given \(OP = 5\;{\rm{cm}}\),radius \(r = 3\;{\rm{cm}}\)
In the right-angled \(\Delta OTP\)
\(O{P^2} = O{T^2} + P{T^ii}\) (past Pythagoras theorem)
\({(5)^2} = {(3)^2} + P{T^2}\)
\( \Rightarrow P{T^2} = {(5)^2} – {(3)^2} = 25 – 9 = sixteen\)
\(\Rightarrow PT = 4\;{\rm{cm}}\)
Hence, the length of the tangent \(PT = four\;{\rm{cm}}.\)
Q.iii. If radii of two concentric circles are \(4\,{\rm{cm}}\) and \(five\,{\rm{cm}}\) respectively, then find the length of the chord of the bigger circle, which is a tangent to the smaller circle.
Ans:
Given, \(OA = 4\;{\rm{cm}},OB = 5\;{\rm{cm}}\)
also, \(OA \bot BC\)
\(O{B^2} = O{A^two} + A{B^2}\)
\({(5)^2} = {(4)^ii} + A{B^2}\)
\( \Rightarrow A{B^two} = {(v)^2} – {(4)^2} = 25 – 16 = 9\)
\( \Rightarrow AB = iii\;{\rm{cm}}\)
Therefore, \(BC = 2AB = 2 \times 3 = 6\;{\rm{cm}}\)
Q.4. \(\Delta ABC\) is circumscribing a circle in the given figure. Notice the length of the side \(BC\).
\(AN = AM = 3\;{\rm{cm}}\) (Tangents drawn from same external betoken are equal)
\(BN = BL = 4\;{\rm{cm}}\)
\(CL = CM = AC – AM = 9 – 3 = 6\;{\rm{cm}}\)
\( \Rightarrow BC = BL + CL = 4 + 6 = 10\;{\rm{cm}}\)
Q.5. In the given figure, ii circles bear upon each other at indicate \(C\). Prove that the common tangent to the circle at \(C\), bisects the common tangent at \(P\) and \(Q\).
Solution: We know that the tangents fatigued from an external point to a circle are equal.
Therefore, \(RP=RC\) and \(RC=RQ\)
\(⇒RP=RQ\)
\(⇒R\) is the midpoint of \(PQ\).
Summary
In this article, we discussed that the tangent to a circle is a line that touches the circle at exactly i betoken. With the aid of real-life applications, one tin hands relate to the concept. We too discussed properties related to the tangent to a circle, the number of tangents drawn from a point lying within the circle, on the circle and outside the circumvolve.
With the assistance of this, one can easily understand the divergence betwixt a chord and a tangent line.
SOLVE QUESTIONS ON TANGENT OF CIRCLE
Frequently Asked Questions
We have provided some ofttimes asked questions almost tangent of a circle here:
Q.one. What is the point of tangency of a circle?
Ans: The betoken of contact of the circle and the tangent line is called the indicate of tangency of a circumvolve.
Q.2. What is/are the number of tangents drawn from a bespeak outside the circumvolve?
Ans: When a indicate lies outside the circle, two tangents can be fatigued from that point to the circle.
Q.iii. What is/are the number of tangents drawn from a signal inside the circumvolve?
Ans: When a point lies inside the circumvolve, no tangent can be drawn.
Q.4. What is a tangent of a circle?
Ans: A tangent to a circle is a line that intersects the circle at simply one signal.
Q.5. Does radius bisect a tangent?
Ans: No, it does non bisect the tangent, merely information technology is perpendicular to the tangent through a point of contact.
We hope this detailed article on tangent of a circle helped you in your studies. If you have whatever doubts or queries on this topic, comment us down below and we will help you lot at the earliest.
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