Class 10th Mathematics CBSE Solution
Exercise 10.1- How many tangents can a circle have?
- Fill in the blanks : (i) A tangent to a circle intersects it in point (s). (ii)…
- A tangent PQ at a point P of a circle of radius 5 cm meets a line through the…
- Draw a circle and two lines parallel to a given line such that one is a tangent…
Exercise 10.2- From a point Q, the length of the tangent to a circle is 24 cm and the distance…
- In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so…
- If tangents PA and PB from a point P to a circle with centre O are inclined to…
- Prove that the tangents drawn at the ends of a diameter of a circle are…
- Prove that the perpendicular at the point of contact to the tangent to a circle…
- The length of a tangent from a point A at distance 5 cm from the centre of the…
- Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord…
- A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove…
- In Fig. 10.13, XY and XY are two parallel tangents to a circle with centre O…
- Prove that the angle between the two tangents drawn from an external point to…
- Prove that the parallelogram circumscribing a circle is a rhombus.…
- A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the…
- Prove that opposite sides of a quadrilateral circumscribing a circle subtend…
- How many tangents can a circle have?
- Fill in the blanks : (i) A tangent to a circle intersects it in point (s). (ii)…
- A tangent PQ at a point P of a circle of radius 5 cm meets a line through the…
- Draw a circle and two lines parallel to a given line such that one is a tangent…
- From a point Q, the length of the tangent to a circle is 24 cm and the distance…
- In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so…
- If tangents PA and PB from a point P to a circle with centre O are inclined to…
- Prove that the tangents drawn at the ends of a diameter of a circle are…
- Prove that the perpendicular at the point of contact to the tangent to a circle…
- The length of a tangent from a point A at distance 5 cm from the centre of the…
- Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord…
- A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove…
- In Fig. 10.13, XY and XY are two parallel tangents to a circle with centre O…
- Prove that the angle between the two tangents drawn from an external point to…
- Prove that the parallelogram circumscribing a circle is a rhombus.…
- A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the…
- Prove that opposite sides of a quadrilateral circumscribing a circle subtend…
Exercise 10.1
Question 1.How many tangents can a circle have?
Answer:Tangent is a line that intersects the circle at one point.
Since, there are infinite number of points on circle. At every point, there is one tangent.
Hence, there are infinite number of tangents in a circle.
Question 2.Fill in the blanks :
(i) A tangent to a circle intersects it in point (s).
(ii) A line intersecting a circle in two points is called a .
(iii) A circle can have parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called.
Answer:(i) A tangent to a circle intersects it in One point.
![](data:image/png;base64,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)
XY is a tangent which intersect the circle at A.
(ii) A line intersecting a circle in two points is called a Secant.
![](data:image/jpeg;base64,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)
(iii) A circle can have Two parallel tangents at the most.
![](data:image/png;base64,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)
AB and CD are two parallel tangents
(iv) The common point of a tangent to a circle and the circle is called the Point of contact.
![](data:image/jpeg;base64,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)
A is point of contact here.
Question 3.A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
A. 12 cm B. 13 cm
C. 8.5 cm D. 119 cm.
Answer:(D)
We know that the line drawn from the centre of the circle to the tangent is perpendicular to the tangent.
OP êž± PQ
By applying Pythagoras theorem in ΔOPQ,
![](data:image/png;base64,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)
OP2 + PQ2 = OQ2
52 + PQ2 =122
PQ2 =144 - 25
PQ =
cm.
Question 4.Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Answer:![](data:image/png;base64,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)
Steps of Construction:
1) Draw a circle with any radius and center O.
2) Choose any point P on the circumference of circle, and draw a line passing through P, Let's name it CD.
3) Draw a line AB parallel to CD, such that AB intersects the circle at two points P and A.
Here,
AB and CD are two parallel lines.
AB intersects the circle at exactly two points, P and Q. Therefore, line AB is the secant of this circle.
CD intersects the circle at exactly one point, R, line CD is the tangent to the circle.
How many tangents can a circle have?
Answer:
Tangent is a line that intersects the circle at one point.
Since, there are infinite number of points on circle. At every point, there is one tangent.
Hence, there are infinite number of tangents in a circle.
Question 2.
Fill in the blanks :
(i) A tangent to a circle intersects it in point (s).
(ii) A line intersecting a circle in two points is called a .
(iii) A circle can have parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called.
Answer:
(i) A tangent to a circle intersects it in One point.
XY is a tangent which intersect the circle at A.
(ii) A line intersecting a circle in two points is called a Secant.
(iii) A circle can have Two parallel tangents at the most.
AB and CD are two parallel tangents
(iv) The common point of a tangent to a circle and the circle is called the Point of contact.
A is point of contact here.
Question 3.
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
A. 12 cm B. 13 cm
C. 8.5 cm D. 119 cm.
Answer:
(D)
We know that the line drawn from the centre of the circle to the tangent is perpendicular to the tangent.
OP êž± PQ
By applying Pythagoras theorem in ΔOPQ,
OP2 + PQ2 = OQ2
52 + PQ2 =122
PQ2 =144 - 25
PQ = cm.
Question 4.
Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Answer:
Steps of Construction:
1) Draw a circle with any radius and center O.
2) Choose any point P on the circumference of circle, and draw a line passing through P, Let's name it CD.
3) Draw a line AB parallel to CD, such that AB intersects the circle at two points P and A.
Here,
AB and CD are two parallel lines.
AB intersects the circle at exactly two points, P and Q. Therefore, line AB is the secant of this circle.
CD intersects the circle at exactly one point, R, line CD is the tangent to the circle.
Exercise 10.2
Question 1.From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
A. 7 cm B. 12 cm
C. 15 cm D. 24.5 cm
Answer:(A)
![](data:image/png;base64,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)
Let O be the centre of the circle.
Given that,
OQ = 25cm and PQ = 24 cm
As the radius is perpendicular to the tangent at the point of contact,
Therefore, OP ⊥ PQ
Applying Pythagoras theorem in ΔOPQ, we obtain
OP2 + PQ2 = OQ2
OP2 + 242 = 252
OP2 = 625 - 576
OP2 = 49
OP = ![](data:image/png;base64,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)
Therefore, the radius of the circle is 7 cm.
Question 2.In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠ POQ = 110°, then ∠ PTQ is equal to
A. 60° B. 70°
C. 80° D. 90°
![](data:image/png;base64,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)
Answer:Given: TP and TQ are tangents.
Therefore, radius drawn to these tangents from centre of the circle will be perpendicular to the tangents.
Thus, OP ⊥ TP and OQ ⊥ TQ
And therefore,
∠OPT = 90°
∠OQT = 90°
In quadrilateral POQT,
Sum of all interior angles = 360°
∠OPT + ∠POQ +∠OQT + ∠PTQ = 360°
90°+ 110° + 90° + ∠PTQ = 360°
∠PTQ = 70°
Question 3.If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to
A. 50° B. 60°
C. 70° D. 80°
Answer:Given: PA and PB are tangents.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ8AAACdCAAAAACPuPU3AAAABGdBTUEAALGOfPtRkwAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAA/MSURBVHja7dxZWFPJngDw+zBP8zAvd+ab+fo+3PnGGe+1vfbY3pletKdbu0fFVlwQBYRmUQRlb0FACIYY1iBr2A37oiyKEBAREhYFEhbZwpqwQwgkhC37VhMWNyQ5S064/c1nPSQnh6Tyo06df9Wp1Knfgd92+t0n3yffJ98n3yffJ98nH7K0MPlb9qlfxjz7TZdfwjckDH3KNQ2mOpWo8YSDGDvfMrNnek0ukWPlW21cSDhfocDMpxi9EdBcGRgxjpFvpUrY7+S9il39m9/nOdbufSRfhk3xNSU+pll9wdBg5uN+nwIAbT8BGx/34dPnLfivMkXYlZ9pqJiXeTAdE56864n23GDjPMewK79Djpzqmy69mPgmi59qfcr84y/VWPl4x+yHBZxZBSa+oSbhOqw/mbaC2fG9FLeGVXABK5ssmWBehpVvwbQU/ObSe76xMyTB+vPiokolE2vjglwFVNuTRs1lj7PHOWNjnHcPE1OTk1MTnPFRztT05OT01DRvnjs1tYStTzMcReoDyml6XkkLs47awe0qrO9vbtmWGC0UUgKJfD82PoocFRenfYiPjUpMTEokx0clk5OyyCF3fL197t7z8/SNaeYM9XGmZ+YEqwosyk++vCgFc5bf3w4PvHfHK5hoecA+wnd7uu0blZqSlJKUnPzuITkpKpFMiklKjo8Pdz596M/7vzh46E9/+eqzg5YmB67444Ki8lsFEplSbaBvI02cimCxWayerp6XITeK+l5vT52vR7kz29Ls9MTMHHeuN8vn6N6TR00cI8rq6mrq8v8aTH/+mN74IgPnc9v2lH00dQ0DH8//sVq80a2UlGZOw85G1ZZyNzT4/v3IIkJglXAzHNh1bJ13r+srk4lEf1wYMbFeaqBPGE3gsMMeaaODIDsSZqhebqfEh7o6hxYNa19URzZv7uXYNrz/Jk0nleR6zZ1AqV02xCcqCGTyI0+ULAPu/fBBGBmsseoeeP3k9nRiqxKXBL7Y3Bo4Xv1RUGTEnj3jmtY4ht6nYQcWq7kxuHYwaJcP3baLZ4uvnbKOHFl6E4Wluf60za3e/R+HU7WYv0Az3+czKJCj9AG5Xbg2BI4uA9b1DshPCyK/sitvZ8292yMrCd06vsMnq3f8jLyXSjj38yO0PtX37hvP/AgzjvYkqaFvdNEHi9u1j1N55e+/lZ9u63C/5sPYJs0nPOOLNEA18/DLKl1fOpIfamddiM6ntCauP01FfHu0QDpLpb2s6Qei5saK2gkpi5EbN/iuOaWHOrp/dIEmK3T3i41PK8lNufEPZbq/Vp3n4ORbJEXhU1NStfGhDW/rR/KqiL4FgFso4HkwQWf2bHiT+kmeYKucWOXO7pUft/2q7ozIMFwAMSI9/EC13ohUeOJyahdyH6i++3CywcyiQSHJJNq4SmTXI5RDtzpAU1h7YOl0aeni5plY6WgZN7NTi6BRKeQymUyuGLKk6w+Z4uITV9qXEfvoXpbmPwb1aKvVCveRC7mgZHIyJX0WDCRkZcUm0ISq9TLuj7dxKePrz3n4f6hQJ3/drzYJXKQ+/s1vHlGnto7i4wPft4HBYMo8GEuIYg31bjYNTZds8gegKk/33hLIozeU6n6zHqEPeJm8+47cgAPEeV5mxiwYIiUtbv0DSRY4KnSvs2tPEYzoXu99NYuHzEdweruZi1+LOpIjrSKxQVd43Wa4Xkg86Qvnoqzr34vhBJAJknnwHCIf2eftqERZBFgx8VI03GCo6ZGzG7vmoi3SYF3TwvQBUHgyaASJr8iHoXzzUR+RHJ+qmbxTNf0ke0MtCDiVKQGY+qRVn18b0MD30Xxz3hy/ucbsRx0LYCr6/jPWwnoW/HiLnCWArQ/IqV9f6oTv6w8JF74NUq1t2kAzH7kvauMf5ATfKIM7IADfB8DTsw4M2L6VPO/5D/co6swC16tIr69VOexeOhIfoF/1GVTD9IHSM7Pbd02XaMMUz/dCJfyvROQDXR4us3B92YcGP6quSgVYxttXqozlA8+PFq7B9L1yf75DJZPm/pGIZCwAoU9CvxAthOfrIZbvMKzY8d//cpNpPB/QEE0r4flYIdkLH+3k3MFleKUY0zfte7MVlo9LCRvdvm+C4KQELxuM6NN+4qb7jAaGT1bjvP3SQxJ2sXrj3JD3LxjNB+qcw5Zg+ABzf8WHO9RlDvGbV1xck6BFlbF88oKfGuD42v/w+MOq0WNN3ArZkheW19uN5QMTXrYDMHxDZx98UM5i0qV3fXVq8qjRfKrWM6RpaB+fHNH/3kvFq/OJfMTfhcYHpEXH8qB9a4+J7zfXbJerH5a6eqGHbxwfUJh68SF94qo77117qYq+rvnwtJfXWiSuKI3iA5XOFEifes6t4N2rXnyQYNvf+Wk2xFHj+KT+v0L6gNw06t2LzOvjH3UXltPjF43jA4VXy6VQPpFF1FsS28tvh3cIZoCRfFPBdhIon4xIenvdHOlA2zkjZWvTkhF8oNquRw7hU6T7N21uaUTWeB3XLtKYKxkCDfa+yRivUQifqjxwq68jobqX68hHPZ90JVWKvU9dt78OwqfpIeRs1bPATKHOnPh0phJ7H+BYRU3q94G10IjNjUU7qFEe7iLWPnGNXQWED+B9Nw9vqzMdIreEeDHGPsA3z4HyEXw2xm1HidHjEJkFm+bNq7D1rbpnSSF8mbjh9VOzw7EOqh1bTjYrFWPrk2WHMCB8VXefrRcg/X+7IXMTds5gXH6akYAHED4GgaIt4vkMF3hTqKZEWPqA0sUXwjdDwmuvMel+lbAGqzTVuCosfSDAhafR61MlXNHGDfI5mDPQmi0uNIgx9BV4Vin1+gDlrDYuBx+G+Zuoih9qNoOhry2ILNPve+y6BECKnQRujoIBOYY+6QOPNf2+Wo9G6WIyCdHsi/YOrHwg3mRJv68jKH+lk1ioRJDnar5bEVa+ZCupft94Ipn3Al+LZNLAEvP25aY1bHwVd8ZUen3L2Tasx/cGkE0/m0hLfNuv7fq3IgN8LUEPV/T6NDVfvCJ5SxBmKxIq1VuNSfefSgzwccgJPL0+wDjeft0WRc5jNNbG86BJtQG+uazwWf2+Pq/OyzYocl6IO1+0XglZhysM8PEogVP6faP4bIsgNFk3nHeaMbj+LeX7T+j3cSOczmWiynswb32MkHOi1gCfqtZ3VL9vtdDsSC66zBVqsCYcukYzwAdeWbH0+5T1Z4+gPwOz7e/92W3cAN+zvzD0+8DY4c/Xpz9MNRSXShFnX374D3/3H34d6H3Vh9ogfMKD/0Qqv48LjPGxb0U2CXK2rzdu79/v+ee9Keh9td91QPiWrxxwCrI/javNvUqeh1epBTP9LeVl1MSIJPvP/vWr33+eb8zjK0mwjuiTyZSa8RgSRE1SrC1x2UOcV4/S7nocP+qEIz954n3wyz/eGELvq9jTAuGT5TnkbXQP1J2hfTreo1EpZRLJKqM4xefMMXPv0OjcthH29PySSDoY4Ylr02hQzy5+fpAJ4VMzrydvbg38oGOq0kzdw3inE2c9gvChqaXU2o6hsXeDs7wxbVdhrAvt/QYvL/RC+MCimdvmRsc/PtwWG/te1qQ42gdE4O4SwvAhlOpXXTv9bFMf39efXorqFgFNg+cIlE9par+5MXSy7E2hjHQ0PUmKjSEEErxNf3ZLLq7v190Dq/ONFChSnfIEKHyiJ3hd7ZtiZWlhamb9uDi7bfaVxHGU4fmx3pbmGsp9XJDTN/91IyrzBVTwVU8fDV1/jrauUyH3zT4I09V/EdAomWk5jCUA/BwnVHLRsoAd7RyeHmT9zSHzwNj81wu8OaFIAjlleDnx2MYg+zPHBBRTyAej4ud1+OTlFzPZUyTLVZDoUsGgxvtcPGrhhE8tpjU1do5yhXALg+ti0bj+3H7SBMUU/M6wwiVd9S9vH10bfy7mCV9cvBgZHZeblVrF7BtbRvgN8z7mGx2Y7iOHlpD7Gu61KnT5HhyY1L7BLHnp1S/XKltGAbokDLmw0UFtt3ZEcY9KvtPmdImdfDn7q8d7vI/NABbJgJtVRKVHN7qPFKtSFNPbo39a0enL/8y/KNw+kqdSJ1hL0APXzvpyV0Vz5qbTyD+rDDdZ1ukrPMwQzL7w8p0AhG8NuX1olOhOpd22r0LRxg0G4sS6j+/epxow77KPCSiuKgN8gF0/MkDvQdPCVd4qUOj2/T5BAxq/MxsB9XcbDbwDEOX/l3ZvVK3LJ807kd/DzPSiy8BEtKvht6CiSYTQrY0dfEJaYX1R7EZsBfQzw38LnsSPoNunkslkq8LNWtN0rk+z+zxFRQBVt+/9NOSXNr37vuVLnlx4PnGDefXu+8QH76jh+cD86ZJd54nqbSB/f3uTBEHZwt32cewoArg+WY1L8S7zNK0/NgG4PiA677/LvoXsW2z4PhDkucsh8HXIczECX/+vQbvro/z03v1TMNYXiDg9o95F3sh1KyEiX7NnyNru8eR4GzZA5JNTvhzaPV+dExkg84EuK9z4bvHUtu48pD7x0y/yduvo0k5lAKQ+wLf5uWV3fP0XCWPIfaD9kMeutHLSlO+2TZCF51OmWaSrjM9TMTzTBWh8YDX4Et34wLZA0pwGlQ+wf7WQGZs3Q3Blb98He32algtxAiP7Ul0+/kUHtk9ZeuShcUuw254C0PsAL9K1xpjAfru7Y4b4AC/Y5qkReW7XmGqDfGDM4TTdWLzGXzyZOw2EIFq/qdX0hz7j8Dptz+181wGy9aWe/ezXrjQCr/eG1QuAgQ8MOVxrkWM+otDt9osOHnzf1rSMJodbNQvY6lSMi246jwps3+LoxoCccsj5Eg1bX9UPTh06Dwlsn5i/1f4y8d6PMBx0m82ysurS/WcU66/NxNrhWVjx2nAWwfpiApr14YQPLK89WcFCt8i8ZZ+pt1lHtX6dhPrdt9lcgxs7ZRfZjzioP16hW19POkAx23lWMZJEszgOObqIdv0/cR3+crhBQ5ft/haEmjlgJB8ACxkevgndaD89QPGyj52Ffp8h6ye23rH3yWhG0Zosd1PDXEiwJscYtL6jZtzvP22p/Txk03fmunK8r0d0wPu/DFx/UjCUfPx4aNUCXKFaJurBn75G7oX7M7LB62Py63OIDkc84S2lpGwmXbYIyO2cg509Fut3DpQFutywcy2EqO6MbHe7gNCorF4ko3UYrS/KirT6xS+1ICG+pnN4+0+yqsmJkdZH5AQfxwv2+WyEGWO3/qnyZazL0SNe96LKmPUv6hmveweHB9tptY0t5THkhGDbH80iy+eQ54rh+qwq8YpgtrUg2uf8wT1/PXfVw9fP+/KBPV+b2PmFpD/nLAolaHreWK8fq1ya43S1NLW+7mM1lNcxm5sYHf0T0wuom0Jjrr8rxuBS5dP6z/+/ff8HDaA8mQ/mfswAAAAASUVORK5CYII=)
Therefore, the radius drawn to these tangents will be perpendicular to the tangents.
Thus, OA ⊥ PA and OB ⊥ PB
∠OBP = 90°
∠OAP = 90°
In quadrilateral AOBP,
Sum of all interior angles = 360°
∠OAP + ∠APB +∠PBO + ∠BOA = 360° 90° + 80° +90° + ∠BOA = 360°
∠BOA = 100°
In ΔOPB and ΔOPA,
AP = BP (Tangents from a point)
OA = OB (Radii of the circle)
OP = OP (Common side)
Therefore, ΔOPB ≅ ΔOPA (SSS congruence criterion)
And thus, ∠POB = ∠POA
![](data:image/png;base64,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)
Question 4.Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Answer:![](data:image/png;base64,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)
Let AB is diameter of the circle. Two tangents PQ and RS are drawn at points A and B respectively. Radius drawn to these tangents will be perpendicular to the tangents.
Thus, OA ⊥ RS and OB ⊥ PQ
∠OAR = 90°
∠OAS = 90°
∠OBP = 90°
∠OBQ = 90°
It can be observed that
∠OAR = ∠ OBQ (Alternate interior angles)
∠OAS = ∠ OBP (Alternate interior angles)
Since alternate interior angles are equal, lines PQ and RS will be parallel.
Question 5.Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Answer:to Prove: A line drawn perpendicular from P will pass from O.
Given: APB is tangent to circle with centre O.
Let us consider a circle with centre O. Let AB be a tangent which touches the circle at P.
![](data:image/png;base64,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)
Proof:
Assume that the perpendicular to AB at P does not pass through centre O. Let it pass through another point O'. Join OP and O'P.
![](data:image/png;base64,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)
As perpendicular to AB at P passes through O', therefore,
∠ O'PB = 90° ... (1)
O is the centre of the circle and P is the point of contact.
We know the line joining the centre and the point of contact to the tangent of the circle are perpendicular to each other.
∴ ∠ OPB = 90° ... (2)
Comparing equations (1) and (2), we obtain
∠ O'PB = ∠ OPB ... (3)
From the figure, it can be observed that,
∠ O'PB < ∠ OPB ... (4)
Therefore, ∠ O'PB = ∠ OPB is not possible. It is only possible, when the line O'P coincides with OP.
Therefore, the perpendicular to AB through P passes through centre O.
Hence, Proved.
Question 6.The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4cm. Find the radius of the circle.
Answer:![](data:image/png;base64,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)
Let us consider a circle centred at point O.
AB is a tangent drawn on this circle from point A.
Given that,
OA = 5cm and AB = 4 cm
In ΔABO,
OB ⊥ AB (Radius ⊥ tangent at the point of contact)
Applying Pythagoras theorem in ΔABO, we obtain
AB2 + BO2 = OA2
42 + BO2 = 52
16 + BO2 = 25 BO2 = 9
BO = 3
Hence, the radius of the circle is 3 cm.
Question 7.Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Answer:![](data:image/png;base64,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)
Let the two concentric circles be centred at point O. And let PQ be the chord of the larger circle which touches the smaller circle at point A. Therefore, PQ is tangent to the smaller circle.
OA ⊥ PQ (As OA is the radius of the circle)
Applying Pythagoras theorem in ΔOAP, we obtain
OA2 + AP2 = OP2
32 + AP2 = 52
9 + AP2 = 25
AP2 = 16
AP = 4
In ΔOPQ,
Since OA ⊥ PQ,
AP = AQ (Perpendicular from the centre of the circle bisects the chord)
PQ = 2AP = 2 × 4 = 8
Therefore, the length of the chord of the larger circle is 8 cm.
Question 8.A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that
AB + CD = AD + BC
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
To Prove: AB + CD = AD + BC
Proof:
In the given figure, it can be observed that AB touches the circle at point P; BC touches the circle at point Q; CD touches the circle at point R; DA touches the circle at point S.
Then, we can say from a theorem that " Length of tangents drawn from an external point to the circle are same ",
Therefore,
DR = DS (Tangents on the circle from point D) ….. (1)
CR = CQ (Tangents on the circle from point C) …... (2)
BP = BQ (Tangents on the circle from point B) ….... (3)
AP = AS (Tangents on the circle from point A) ….... (4)
Adding all these equations, we obtain
DR + CR + BP + AP = DS + CQ + BQ + AS
(DR + CR) + (BP + AP) = (DS + AS) + (CQ + BQ)
CD + AB = AD + BC
Hence, Proved.
Question 9.In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.
![](data:image/png;base64,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)
Answer:Let us join point O to C.
![](data:image/png;base64,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)
In ΔOPA and ΔOCA,
OP = OC (Radius of the same circle)
AP = AC (Tangents from point A)
AO = AO (Common side)
ΔOPA ≅ ΔOCA (SSS congruence criterion)
∠POA = ∠COA … (i)
Similarly, ΔOQB ≅ ΔOCB
∠QOB = ∠COB … (ii)
Since POQ is a diameter of the circle, it is a straight line.
Therefore, ∠POA + ∠COA + ∠COB + ∠QOB = 180°
From equations (i) and (ii), it can be observed that 2∠COA + 2 ∠COB = 180°
∠ COA + ∠ COB = 180° / 2
∠COA + ∠COB = 90°
∠AOB = 90°
Hence Proved.
Question 10.Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
Answer:![](data:image/png;base64,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)
To Prove: ∠ APB + ∠ BOA = 180°
Proof:
Let us consider a circle centred at point O.
Let P be an external point from which two tangents PA and PB are drawn to the circle which touches the circle at point A and B respectively and AB is the line segment, joining point of contacts A and B together such that it subtends
∠ AOB at centre O of the circle.
It can be observed that
OA ⊥ PA (radius of circle is always perpendicular to tangent)
Therefore, ∠ OAP = 90°
Similarly, OB ⊥ PB
∠ OBP = 90°
In quadrilateral OAPB,
Sum of all interior angles = 360°
∠ OAP +∠ APB+∠ PBO +∠ BOA = 360°
90° + ∠ APB + 90° + ∠ BOA = 360°
∠ APB + ∠ BOA = 180°
Hence, it can be observed that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
Question 11.Prove that the parallelogram circumscribing a circle is a rhombus.
Answer:Since ABCD is a parallelogram,
AB = CD ...(1)
BC = AD ...(2)
![](data:image/png;base64,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)
It can be observed that
DR = DS (Tangents on the circle from point D)
CR = CQ (Tangents on the circle from point C)
BP = BQ (Tangents on the circle from point B)
AP = AS (Tangents on the circle from point A)
Adding all these equations, we obtain
DR + CR + BP + AP = DS + CQ + BQ + AS
(DR + CR) + (BP + AP) = (DS + AS) + (CQ + BQ)
CD + AB = AD + BC
On putting the values of equations (1) and (2) in this equation, we obtain 2AB = 2BC
AB = BC ...(3)
Comparing equations (1), (2), and (3), we obtain
AB = BC = CD = DA
Hence, ABCD is a rhombus.
Question 12.A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.
Answer:![](data:image/png;base64,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)
To Find: AB and AC
Let the given circle touch the sides AB and AC of the triangle at point E and F respectively and the length of the line segment AF be x.
In ∆ABC,
From the theorem which states that the lengths of two tangents drawn from an external point to a circle are equal.
So,
CF = CD = 6cm (Tangents on the circle from point C)
BE = BD = 8cm (Tangents on the circle from point B)
AE = AF = x (Tangents on the circle from point A)
AB = AE + EB = x + 8
BC = BD + DC = 8 + 6 = 14
CA = CF + FA = 6 + x
By heron's formula,
![](data:image/png;base64,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)
where, a, b and c are sides of triangle and s is semi perimeter.
2S = AB + BC + CA
= x + 8 + 14 + 6 + x
= 28 + 2x
S = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
=√(14 + x)(x)(8)(6)
![](data:image/png;base64,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)
Also, area of triangle = ![](data:image/png;base64,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)
And from the given figure it is clear that
Area of ΔABC = Area of ΔOBC + Area of ΔOCA + Area of ΔOAB
Area of ΔOBC =![](data:image/png;base64,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)
Area of ΔOCA = ![](data:image/png;base64,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)
Area of ΔOAB = ![](data:image/png;base64,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)
Area of ΔABC = Area of ΔOBC + Area of ΔOCA + Area of ΔOAB
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMsAAAAeCAMAAAC11hqdAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OjoAOjo6OjqQOma2OpDbZgAAZgA6ZgBmZjqQZmZmZrb/kDoAkDo6kDpmkLaQkNv/tmYAtmY6ttuQtv//25A625Bm27Zm2////7Zm/9uQ//+2///b5/zKLQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACqElEQVRYR+1X21LjMAxtSgmwEGA3sDVgqJP8/z9iSb5IjpIy0JaBSR4Y1kaXI50jeVer5VsqsFRgqcBvrICtjvydrmjDw/Z0wY4cqfuzO3KE07l352Use/ZyoPBmc9o6mX8y8aG9+TKSvqkqLFFXF94V169f4Lhdc+P+DprgvPzpOEPpLtLfSZO9UPvb7cqRO+ZkbNbVMHViOraa5YOWQ1cLLCgX44+GFs9dcOhLm/6uMMlJDe1k2SMIO0Mz3rWullwvXWs59M1fgcUCowz8sJVPbGjJpYeUClGafAgL+t3TGIalqwtmF1jUHMxN6H7I6CkRyUBHmPuEJZoMbeUDQg/jJwKK6zRR5uSXg8USTpZJy8FtdgILyQV1gmRglxFLMnna+glnOasEFn5tM2HMaE6mhDMWB5wQn+yLlgNTJVmG+sHgmcDCTdz6/zWPWBAhXYNXF7KLc9mk1wXT4VWQvjl79nOAS0u41nLAhvvS5+Lm37oaOMZ2C/VFmPRNHjVKbvEaKlPFSs/sGPCNI8fT0+NI8i9dqznA6MUvdpQ/YByoYcQxYVKSv+iLqo19+xJUHwiOig0fdz2dA6XbN4CHP2BwlCh6YSpyggbQNEHz8hrTinpR+oj3fXMesfAVMhr3MTMRBA/N+hLYSXKheqL+tDmWEPb3b3KdyYCj6+y7EHb4J64CLCF1ZKovpGySmQwS0GPq9IAZWu+K+syGI6tSkM7ji4fq2CKQM3l0Db7nFj+taFoFMC2A5BrH8EzLwZMviHqzGx6JoCjWtCQJHuoqnKEJOPPHfMRyLMo1pjCz9zEs3eNzZnLcI2smcohAlff+IZ6WjFGz7zGdeZ859aKLDw1uflAwH3knfyb3kY1dX6nP7Z/4/xe++Q5SnO90wl5N35nGEvsXVeAd+N48uHU/8BsAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒![](data:image/png;base64,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)
⇒ 3x = 14 + x
⇒ 2x = 14
⇒ x = 7
Therefore, x = 7
Hence,
AB = x + 8 = 7 + 8 = 15 cm
CA = 6 + x = 6 + 7 = 13 cm
Hence, AB = 15 cm and AC = 13 cm
Question 13.Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Answer:![](data:image/png;base64,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)
To Prove : ∠ AOB + ∠ COD = 180°, ∠ BOC + ∠ DOA = 180°
Given: ABCD is circumscribing the circle.
Proof:
Let ABCD be a quadrilateral circumscribing a circle centred at O such that it touches the circle at point P, Q, R, S.
join the vertices of the quadrilateral ABCD to the centre of the circle.
Consider ΔOAP and ΔOAS,
AP = AS (Tangents from the same point)
OP = OS (Radii of the same circle)
OA = OA (Common side)
ΔOAP ≅ ΔOAS (SSS congruence criterion)
And thus, ∠ POA = ∠ AOS
∠1 = ∠ 8 Similarly,
∠2 = ∠ 3
∠4 = ∠ 5
∠6 = ∠ 7
∠1 + ∠ 2 + ∠ 3 + ∠ 4 + ∠ 5 + ∠ 6 + ∠ 7 + ∠ 8 = 360°
(∠ 1 + ∠ 8) + (∠ 2 + ∠ 3) + (∠ 4 + ∠ 5) + (∠ 6 + ∠ 7) = 360°
2∠ 1 + 2∠ 2 + 2∠ 5 + 2∠ 6 = 360°
2(∠ 1 + ∠ 2) + 2(∠ 5 + ∠ 6) = 360°
(∠ 1 + ∠ 2) + (∠ 5 + ∠ 6) = 180°
∠ AOB + ∠ COD = 180°
Similarly, we can prove that ∠ BOC + ∠ DOA = 180°
Hence, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
A. 7 cm B. 12 cm
C. 15 cm D. 24.5 cm
Answer:
(A)
Let O be the centre of the circle.
Given that,
OQ = 25cm and PQ = 24 cm
As the radius is perpendicular to the tangent at the point of contact,
Therefore, OP ⊥ PQ
Applying Pythagoras theorem in ΔOPQ, we obtain
OP2 + PQ2 = OQ2
OP2 + 242 = 252
OP2 = 625 - 576
OP2 = 49
OP =
Therefore, the radius of the circle is 7 cm.
Question 2.
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠ POQ = 110°, then ∠ PTQ is equal to
A. 60° B. 70°
C. 80° D. 90°
Answer:
Given: TP and TQ are tangents.
Therefore, radius drawn to these tangents from centre of the circle will be perpendicular to the tangents.
Thus, OP ⊥ TP and OQ ⊥ TQ
And therefore,∠OPT = 90°
∠OQT = 90°
In quadrilateral POQT,
Sum of all interior angles = 360°
∠OPT + ∠POQ +∠OQT + ∠PTQ = 360°
90°+ 110° + 90° + ∠PTQ = 360°
∠PTQ = 70°
Question 3.
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to
A. 50° B. 60°
C. 70° D. 80°
Answer:
Given: PA and PB are tangents.
Therefore, the radius drawn to these tangents will be perpendicular to the tangents.
Thus, OA ⊥ PA and OB ⊥ PB
∠OBP = 90°
∠OAP = 90°
In quadrilateral AOBP,
Sum of all interior angles = 360°
∠OAP + ∠APB +∠PBO + ∠BOA = 360° 90° + 80° +90° + ∠BOA = 360°
∠BOA = 100°
In ΔOPB and ΔOPA,
AP = BP (Tangents from a point)
OA = OB (Radii of the circle)
OP = OP (Common side)
Therefore, ΔOPB ≅ ΔOPA (SSS congruence criterion)
And thus, ∠POB = ∠POA
Question 4.
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Answer:
Let AB is diameter of the circle. Two tangents PQ and RS are drawn at points A and B respectively. Radius drawn to these tangents will be perpendicular to the tangents.
Thus, OA ⊥ RS and OB ⊥ PQ
∠OAR = 90°
∠OAS = 90°
∠OBP = 90°
∠OBQ = 90°
It can be observed that
∠OAR = ∠ OBQ (Alternate interior angles)
∠OAS = ∠ OBP (Alternate interior angles)
Since alternate interior angles are equal, lines PQ and RS will be parallel.
Question 5.
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Answer:
to Prove: A line drawn perpendicular from P will pass from O.
Given: APB is tangent to circle with centre O.
Let us consider a circle with centre O. Let AB be a tangent which touches the circle at P.
Assume that the perpendicular to AB at P does not pass through centre O. Let it pass through another point O'. Join OP and O'P.
As perpendicular to AB at P passes through O', therefore,
∠ O'PB = 90° ... (1)
O is the centre of the circle and P is the point of contact.
We know the line joining the centre and the point of contact to the tangent of the circle are perpendicular to each other.
∴ ∠ OPB = 90° ... (2)
Comparing equations (1) and (2), we obtain
∠ O'PB = ∠ OPB ... (3)
From the figure, it can be observed that,
∠ O'PB < ∠ OPB ... (4)
Therefore, ∠ O'PB = ∠ OPB is not possible. It is only possible, when the line O'P coincides with OP.
Therefore, the perpendicular to AB through P passes through centre O.
Hence, Proved.
Question 6.
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4cm. Find the radius of the circle.
Answer:
Let us consider a circle centred at point O.
AB is a tangent drawn on this circle from point A.
Given that,
OA = 5cm and AB = 4 cm
In ΔABO,
OB ⊥ AB (Radius ⊥ tangent at the point of contact)
Applying Pythagoras theorem in ΔABO, we obtain
AB2 + BO2 = OA2
42 + BO2 = 52
16 + BO2 = 25 BO2 = 9
BO = 3
Hence, the radius of the circle is 3 cm.
Question 7.
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Answer:
Let the two concentric circles be centred at point O. And let PQ be the chord of the larger circle which touches the smaller circle at point A. Therefore, PQ is tangent to the smaller circle.
OA ⊥ PQ (As OA is the radius of the circle)
Applying Pythagoras theorem in ΔOAP, we obtain
OA2 + AP2 = OP2
32 + AP2 = 52
9 + AP2 = 25
AP2 = 16
AP = 4
In ΔOPQ,
Since OA ⊥ PQ,
AP = AQ (Perpendicular from the centre of the circle bisects the chord)
PQ = 2AP = 2 × 4 = 8
Therefore, the length of the chord of the larger circle is 8 cm.
Question 8.
A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that
AB + CD = AD + BC
Answer:
To Prove: AB + CD = AD + BC
Proof:
In the given figure, it can be observed that AB touches the circle at point P; BC touches the circle at point Q; CD touches the circle at point R; DA touches the circle at point S.
Then, we can say from a theorem that " Length of tangents drawn from an external point to the circle are same ",
Therefore,
DR = DS (Tangents on the circle from point D) ….. (1)
CR = CQ (Tangents on the circle from point C) …... (2)
BP = BQ (Tangents on the circle from point B) ….... (3)
AP = AS (Tangents on the circle from point A) ….... (4)
Adding all these equations, we obtain
DR + CR + BP + AP = DS + CQ + BQ + AS
(DR + CR) + (BP + AP) = (DS + AS) + (CQ + BQ)
CD + AB = AD + BC
Hence, Proved.
Question 9.
In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.
Answer:
Let us join point O to C.
In ΔOPA and ΔOCA,
OP = OC (Radius of the same circle)
AP = AC (Tangents from point A)
AO = AO (Common side)
ΔOPA ≅ ΔOCA (SSS congruence criterion)
∠POA = ∠COA … (i)Similarly, ΔOQB ≅ ΔOCB
∠QOB = ∠COB … (ii)
Since POQ is a diameter of the circle, it is a straight line.
Therefore, ∠POA + ∠COA + ∠COB + ∠QOB = 180°
From equations (i) and (ii), it can be observed that 2∠COA + 2 ∠COB = 180°
∠ COA + ∠ COB = 180° / 2∠COA + ∠COB = 90°
∠AOB = 90°
Hence Proved.Question 10.
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
Answer:
To Prove: ∠ APB + ∠ BOA = 180°
Let us consider a circle centred at point O.
Let P be an external point from which two tangents PA and PB are drawn to the circle which touches the circle at point A and B respectively and AB is the line segment, joining point of contacts A and B together such that it subtends
∠ AOB at centre O of the circle.
It can be observed that
OA ⊥ PA (radius of circle is always perpendicular to tangent)
Therefore, ∠ OAP = 90°
Similarly, OB ⊥ PB
∠ OBP = 90°
In quadrilateral OAPB,
Sum of all interior angles = 360°
∠ OAP +∠ APB+∠ PBO +∠ BOA = 360°
90° + ∠ APB + 90° + ∠ BOA = 360°
∠ APB + ∠ BOA = 180°
Hence, it can be observed that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
Question 11.
Prove that the parallelogram circumscribing a circle is a rhombus.
Answer:
Since ABCD is a parallelogram,
AB = CD ...(1)
BC = AD ...(2)
It can be observed that
DR = DS (Tangents on the circle from point D)
CR = CQ (Tangents on the circle from point C)
BP = BQ (Tangents on the circle from point B)
AP = AS (Tangents on the circle from point A)
Adding all these equations, we obtain
DR + CR + BP + AP = DS + CQ + BQ + AS
(DR + CR) + (BP + AP) = (DS + AS) + (CQ + BQ)
CD + AB = AD + BC
On putting the values of equations (1) and (2) in this equation, we obtain 2AB = 2BC
AB = BC ...(3)
Comparing equations (1), (2), and (3), we obtain
AB = BC = CD = DA
Hence, ABCD is a rhombus.
Question 12.
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.
Answer:
Let the given circle touch the sides AB and AC of the triangle at point E and F respectively and the length of the line segment AF be x.
In ∆ABC,
From the theorem which states that the lengths of two tangents drawn from an external point to a circle are equal.So,
CF = CD = 6cm (Tangents on the circle from point C)
BE = BD = 8cm (Tangents on the circle from point B)
AE = AF = x (Tangents on the circle from point A)
AB = AE + EB = x + 8
BC = BD + DC = 8 + 6 = 14
CA = CF + FA = 6 + x
where, a, b and c are sides of triangle and s is semi perimeter.
2S = AB + BC + CA
= x + 8 + 14 + 6 + x
= 28 + 2x
S =
=√(14 + x)(x)(8)(6)
Also, area of triangle =
And from the given figure it is clear that
Area of ΔABC = Area of ΔOBC + Area of ΔOCA + Area of ΔOAB
Area of ΔOBC =
Area of ΔOCA =
Area of ΔOAB =
Area of ΔABC = Area of ΔOBC + Area of ΔOCA + Area of ΔOAB
⇒
⇒
⇒
⇒
⇒ 3x = 14 + x
⇒ 2x = 14
⇒ x = 7
Therefore, x = 7
Hence,
AB = x + 8 = 7 + 8 = 15 cm
CA = 6 + x = 6 + 7 = 13 cm
Hence, AB = 15 cm and AC = 13 cm
Question 13.
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Answer:
Given: ABCD is circumscribing the circle.
Proof:
Let ABCD be a quadrilateral circumscribing a circle centred at O such that it touches the circle at point P, Q, R, S.
join the vertices of the quadrilateral ABCD to the centre of the circle.
Consider ΔOAP and ΔOAS,
AP = AS (Tangents from the same point)
OP = OS (Radii of the same circle)
OA = OA (Common side)
ΔOAP ≅ ΔOAS (SSS congruence criterion)
And thus, ∠ POA = ∠ AOS
∠1 = ∠ 8 Similarly,
∠2 = ∠ 3
∠4 = ∠ 5
∠6 = ∠ 7
∠1 + ∠ 2 + ∠ 3 + ∠ 4 + ∠ 5 + ∠ 6 + ∠ 7 + ∠ 8 = 360°
(∠ 1 + ∠ 8) + (∠ 2 + ∠ 3) + (∠ 4 + ∠ 5) + (∠ 6 + ∠ 7) = 360°
2∠ 1 + 2∠ 2 + 2∠ 5 + 2∠ 6 = 360°
2(∠ 1 + ∠ 2) + 2(∠ 5 + ∠ 6) = 360°
(∠ 1 + ∠ 2) + (∠ 5 + ∠ 6) = 180°
∠ AOB + ∠ COD = 180°
Similarly, we can prove that ∠ BOC + ∠ DOA = 180°
Hence, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.