Q. 91. ABC is a right triangle, right angled at C and AC = √3 BC. Find the measures of the angles A and B of ∆ABC.
Q. 92. In a triangle ABC, AD is a median. Prove that: AB2 + AC2 = 2AD2 + 2 DC2.
Q. 93. If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.
Q. 94. In a ∆ PQR, PR = RQ and PQ2 – QR2 = PR2. Determine all the three angles of the triangle.
Q. 95. Prove that the line joining the mid-points of two parallel chords of a circle passes through the centre of the circle.
Q. 96. Prove that the line joining the mid-points of two equal chords of a circle makes equal angles with these chords.
Q. 97. If two non-parallel sides of a trapezium are equal, then prove that it is cyclic.
Q. 98. Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.
Q. 99. Two circles of radii 10 cm and 8 cm intersect at two points and the length of common chord is 12 cm. Find the distance between their centres.
Q. 100. In a circle with centre O, AB is an arc and C is any point its major arc such that measures of the angles CAO and CBO are 320 and 450 respectively. Find the measure of the angle subtended by the arc AB at the centre.
Q. 101. AB and CD are two equal chords of a circle whose centre is O. When produced, these chords meet at a point E outside the circle. Prove that : EB= ED.
Q. 102. In an isosceles triangle ABC with AB = AC, a circle passing through B and C intersects the sides AB and AC at D and E respectively. Prove that: (i) DE // BC. (ii) AD × AC = AE × AB.
Q. 103. AC and BD are chords of a circle that bisect each other. Prove that AC and BD are diameters and ABCD is a rectangle.
Q. 104. The bisectors of the opposite angles A and C of a cyclic quadrilateral ABCD intersect the circle at the points E and F respectively. Prove that EF is a diameter of the circle.
Q. 105. 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. 106. 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. 107. In two concentric circles, prove that all the chords of the outer circle which touch the inner circle are of equal lengths.
Q. 108. 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. 109. 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. 110. 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. 111. 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. 112. 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. 113. 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. 114. 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.
Q. 115. Construct the circumcircle of a right-triangle whose two smaller sides are 4 cm and 5 cm. Also measure the length of the third side of the triangle and the radius of the circumcircle.
Q. 116. Construct a pair of tangents to a circle of radius 6 cm which are inclined at right angles to each other.
Q. 117. Construct an isosceles triangle whose base and the altitude are in the ratio 2:1 and the sum of them is 12 cm. Draw its incircle and measure its radius.
Q. 118. Construct a right-triangle whose larger side is 13 cm and one of the smaller sides is 5 cm. Also draw its circumcircle and measure the length of the third side and the radius of the circumcircle.
Q. 119. Construct a triangle ABC in which AC = 6 cm, angle B = 300 and altitude BD = 5cm.
Q. 120. Construct a cyclic quadrilateral ABCD in which AC = 6 cm, BC = 4 cm, CD = 3.5 cm and angle B is right angle.
