Angle Of Tangent To Circle
A tangent to a circle can exist defined as a line that touches the circle exactly at one point. The point of contact is the indicate at which the tangent touches the circle. Depending on the position of this signal, nosotros can consider that the number tangents than tin exist fatigued to a circle. Atangent is defined as a straight line that intersects the circle exactly at one point. Here, we will be discussing more almost the formulas, and theorems of the length of tangent on a circle along with some of import questions.
Key Terms: Tangent, Tangent of Circle, Tangent Theorems, Pythagoras Theorems, Tangent Formulas, Circle Formulas
What is Tangent to a Circle?
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Tangent to a circle is divers as the line that touches the circumvolve only at one indicate. In that location cannot exist more than one tangent at a point to circumvolve. Point of tangency can be defined as the point at which tangent meets the circle.
Tangent to a Circle
The above figure has a circle with centre O. Where the tangent is fatigued to a circle through point C. A signal D is taken on tangent AB other than C and join OD. The point D which is taken on the tangent AB lies outside the circle because; if point D lies inside the circle, then AB will exist a secant to the circle and it will not exist a tangent as mentioned before.
Tangent Equation
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These are the following equations of the tangent:
- The tangent to a circle equation x2 + yii = atwo at (x1, yone) is xx1 + yy1 = a2
- The tangent to a circle equation xii + y2 + 2gx + 2fy + c = 0 at (x1, y1) is xx1 + yy1 + g(x+ten1) + f(y +y1) + c = 0
- The tangent to a circumvolve equation xtwo + y2 = a2 at (a cos θ, a sin θ) is x cos θ + y sin θ = a
- The tangent to a circle equation xtwo + y2 = a2 for a line y = mx + c is y = mx ± a √[1+ grandtwo]
Tangent Theorems
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Theorem 1
A radius which is obtained by joining the heart and the point oftangency and this tangent at a signal on a circumvolve is at right angles to the radius obtained. To understand the argument, follow the diagram beneath: Hither AB⊥OP
Theorem ii
It states that from one external betoken, two tangents are drawn to a circle and so they have equal tangentsegments. The meaning of tangent segment is the line joining to the external point and the betoken of tangency. To understand the statement, follow the diagram beneath: Hither, AC = BC.
Tangent of Circle Formula
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Suppose a betoken P lies outside the circle.
From that point P, we draw ii tangents to the circle meeting at point Eastward and F. Now let a secant is drawn from P to intersect the circumvolve at G and H. PS is the tangent line from point P to S. At present, the formula for tangent and secant of the circle could exist given as:
PH PS = PS PG
PS2 = PH. PG
Read More: Circumvolve Formulas
Things to Remember
- At a single indicate tangent always touches the circle.
- At the indicate of tangency, it is perpendicular to the radius of the circumvolve.
- A tangent can never cross a circle, which means that it cannot laissez passer through a circle.
- At 2 points tangents never intersects the circle.
- The length of tangents from an external point to a circle are always equal.
- The line of thetangent is ever perpendicular the radius of a circle.
Sample Questions
Ques. Two concentric circles are of radii 10 cm and half dozen cm. Notice the length of the chord of the larger circle which touches the smaller circle? (4 marks)
Ans. In the given figure, MP is the chord of the larger circle, which touches the smaller circle at N.
Nosotros take given, OP = OM = 10cm [Radii of larger circumvolve] and ON = half dozen cm [Radii of smaller circle]
Since M is the tangent to the smaller circle. So ON ⊥ MP (By Theorem)
In Δ OMN and Δ OPN,
∠OMP = ∠ OPM [ Each of 90°]
ON = ON [ Common]
OM = OP [ Radii of the same circle]
Therefore Δ OMN ≅ Δ OPN
MN = NP [CPCT]
In Δ OMN,
MN2 = OMii – ONii = (10)two – (6)ii = 64
MN = √64 = eight cm
MP = 2 MN = 16cm
Ques. Find the length of the tangent in the circle shown below? (3 marks)
Ans. The above diagram has 1 tangent and one secant.
Given us the post-obit lengths:
PQ = 10 cm and QR = twenty cm,
Therefore, PR = PQ + QR = (10 + twenty) cm= 30 cm.
SRii = PR × RQ
SR2 = 30 × 20
SRtwo = 600 cm
√SR = √600
SR = 24.4 cm
Then, the length of the tangent is 24.4 cm.
Ques. If tangents PA and Pb from a signal P to a circumvolve with centre O are inclined to each other at angle of 80°, so ∠ POA is equal to? (iii marks)
Ans.
∠PAO = ∠ PBO [Each of 90°]
OA = OB [Radii of the circle]
PA = PB [Both are tangents]
Δ POA ≅ Δ POB [By SAS congruence]
∠APO = ∠ BPO [CPCT]
∠APO =\(\frac{1}{2}\) ∠APB =\(\frac{i}{2}\) × 80° = 40°
In Δ POA, ∠APO + ∠POA + ∠OAP = 180°
twoscore° + ∠POA + ninety° = 180°
∠POA = fifty°
Ques. In figure, if TP and TQ are the ii tangents to a circle with heart O and so that ∠ POQ = 110°, and then ∠ PTQ is equal to? (three marks)
Ans. ∠OPT = 90°
∠OQT = ninety°
∠APO = 110°
TPOQ is a quadrilateral
Therefore, ∠PTQ + ∠POQ = 180°
∠PTQ + 110° = 180°
∠PTQ = 70°
Ques. If the tangent at a indicate P to a circle with middle O cuts a line through O at Grand such that OS = 5 cm and OP = sixteen cm. Find the value of PS? (iii marks)
Ans. OS is a line, OP ⊥ tangent MN
In right ΔOPS,
Bone² = OP² + PS² (Pythagoras Theorem)
=> (five)² = (4)² + PS²
=> 25 = PS² + 16
=> PS² = 25 – xvi = ix
=> PS² = (3)²
PS = 3 cm
Hence the value of PS = 3 cm
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Source: https://collegedunia.com/exams/length-of-tangent-on-a-circle-formula-and-theorems-mathematics-articleid-6239
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