DOWNLOAD MOBILE APPLICATION TO LEARN MORE: CIRCLE CLASS 10 ICSE NOTES
DOWNLOAD MOBILE APPLICATION TO LEARN MORE: CIRCLE CLASS 10 ICSE NOTES
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CIRCLE CLASS 10 ICSE NOTES
DOWNLOAD MOBILE APPLICATION TO LEARN MORE: CIRCLE CLASS 10 ICSE NOTES
[angle subtended by an arc at the centre is twice the angle subtended at the remaining part of the circumference of the circle]
We know that, exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
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4. If the sides of a parallelogram touch a circle, then prove that the parallelogram is a rhombus.
Sol. Given A parallelogram ABCD in which a circle touches the sides at P, Q, R and S respectively.
To prove ABCD is a rhombus.
Proof Since, tangents from an external point to a circle are equal in length.
Similarly BP = BQ, CR = CQ
and DR = DS
On adding these, we get
AP + BP + CR + DR = AS + BQ + CQ + DS
(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)
AB + CD = AD + BC
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But ABCD is a parallelogram.
2AB = 2BC
AB = BC
AB = BC = CD = AD
Hence, ABCD is a rhombus. Hence proved.
5. Find the length of the unknown part of the chord in the given figures.
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Sol. We know that, when two chords of a circle intersect internally or externally, then the products of the lengths of segments are equal.
Sol. Given AB = 7 cm and BC = 9 cm.
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7. In the figure given below, ‘O’ is the centre of the circle. If QR = OP and
Sol.
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9. Two circles with centres O and O’ touch each other externally at point D. if A is a point on the common tangent through D, then prove that the tangents AB and AC are equal.
Sol. Given Two circles with centres O and O’ touch each other externally at point D. AB and AC are the tangents to the circles with centres O and O’, respectively from point A, which lie on the common tangent AD.
To prove AB = AC
Proof ab and AD are the tangents to the circle (with centre O) from point A.
[ length of tangents drawn from an external point to a circle are equal]
Again, AC and AD are tangents to the circle
(with centre O’) from point A.
AC = AD
[ length of tangents drawn from an external point to a circle are equal]
From Eqs. (i) and (ii), we get AB = AC Hence proved.
10. If a, b and c are the sides of a right angled triangle, where c is hypotenuse, then prove that the radius r of the circle which touches the sides of the triangle is given
AF = AE = AC – EC = b – r and BF = a – r
Now, AB = c
AF + BF = (b – r) + (a – r)
AB = b – r + a – r
c = a + b – 2r
2r = a + b – c
11. Prove that, in a cyclic quadrilateral, if one pair of opposite sides is equal, the other pair is parallel.
Sol. Join O to A, B, C, D and join A to C.
AD = BC
[
Sol. (i) The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
(iv)Angle at centre = 2 X Angle at any point on the remaining part of circle
(v)
and x = y = 50 degree
(vi) We know that angle subtended by an arc at the centre is double the angle subtended by it at any other point on the remaining part of the circle.
13. In the given figure, from an external point P, a tangent PT and a line segment PAB drawn to a circle with centre O. ON is perpendicular on the chord AB.
Prove that
(i) PA
(ii) PN2 – AN2 = OP2 – OT2
(iii) PA
Sol. Given PT is a tangent drawn from an external point P and a line segment PAB is drawn to a circle with centre O. ON is perpendicular on the chord AB.
To prove
(i) PA
(ii) PN2– AN2 = OP2 – OT2
(iii) PA
(i) BD is a diameter of the circle.
(ii) ABC is an isosceles triangle.
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ALSO VISIT :
10th ICSE
| Also Learn: | Attempt theQuiz on Mathematics : | Attempt theQuiz on Physics: | Attempt theQuiz on Biology: | Attempt theQuiz on Chemistry: |
| 1.Surface Areas and Volumes 2. ELECTRICITY 3. METALS AND NON METALS 4. ATOMS AND MOLECULES 5. GST 6. PHOTOSYNTHESIS IN HIGHER PLANTS 7. ABSORPTION BY ROOTS 8. QUADRATIC EQUATIONS 9. LINEAR INEQUATIONS 10. RECURRING DEPOSIT 11. CIRCULAR MOTION 12. MCQ WORK POWER ENERGY 13. ACIDS BASES AND SALTS 14. MCQ ON ACIDS BASES AND SALTS 15. ROTATIONAL MOTION MASS 2 16. ROTATIONAL MOTION MASS 1 17. MCQ ON PERIODIC TABLE 2 18. MCQ ON PERIODIC TABLE 1 19. CELL CYCLE AND CELL DIVISION 20.REFLECTION CLASS 10 ICSE NOTES 21.EQUATION OF LINE CLASS 10 ICSE NOTES | QUIZ ON SURFACE AREAS AND VOLUMES- MATHEMATICS QUIZ ON SECTION AND MID-POINT | QUIZ ON ELECTRICITY- PHYSICS | QUIZ 1 ON METALS AND NON-METALS – CHEMISTRYQUIZ 2 ON METALS AND NON-METALS – CHEMISTRY |