CBSE Guess > Papers > Important Questions > Class X > Maths > Mathematics By Mr. Sanjay Kumar

Chapter - 10

Q. 106. ABCD is a cyclic quadrilateral in which AE is drawn parallel to CD and BA is produced to a point F. If the measures of the angles ABC and FAE are 85⁰ and 35⁰, respectively. Find the measures of the angles BCD and BAD.

Q. 107. If PA and PB are two tangents drawn from a point P to a circle with centre O touching it at A and B, prove that OP is the perpendicular bisector of AB.

Q. 108. In two concentric circles, prove that all the chords of the outer circle which touch the inner circle are of equal lengths.

Q. 109. If AB and CD are two chords of a circle which when produced meet at point P such that AP = CP. Prove that: AB = CD.

Q. 110. ABC is a triangle in which AB = AC. A circle through B touches AC at D and intersects AB at P. If D is the mid-point of AC, then prove that: AB = 4 AP.

Q. 111. The diagonals of a parallelogram ABCD intersect each other at point E. Show that the circumcircles of ∆ADE and ∆BCE touch each other at point E.

Q. 112. AB is a diameter and AC is a chord of a circle such that angle BAC is 30˚. The tangent at C intersects AB produced in a point D. Prove that: BC = BD.

Q. 113. AP is a tangent to a circle at P and ABC is a secant intersecting the circle at points B and C. If PD is the bisector of the angle BPC which meets the chord BC at point D, prove that : 2 ∠ BPD = ∠ ABP – ∠ APB.

Q. 114. A circle is inscribed in a ∆ABC touching the sides AB, BC and CA at the points D, E and F respectively. Find the lengths of AD, BE and CF.

Q. 115. Two circles intersect each other at two points A and B. At A, tangents AP and AQ to the two circles are drawn which intersect the other circles at the points P and Q respectively. Prove that AB is the bisector of the angle PBQ.