64. Two equal circles are placed so that the transverse common tangent is equal to the radius. Show that the tangent from the centre of one circle to the other equals the diameter of each circle. 65. Construct a ▲ having its medians respectively equal to three given st. lines. 66. Construct a ▲ given one side and the lengths of the medians drawn from the ends of that side. 67. Construct a ▲ given one side, the median drawn to the middle point of that side, and a median drawn from one end of that side. - b = 3 cm., and 70. Construct a ▲ having a = 7 cm., c LC-LB = 28°. 71. If a st. line be drawn in any direction from one vertex (f a ||gm, the to it from the opposite vertex equals the sum or difference of the 1s to it from the two remaining vertices. 72. PQ is a chord of a circle to the diameter LM, and E is any point in LM. If PE, QE meet the circumference in S, R respectively, show that PS QR; and that RS LM. = = 73. P is any point in a diameter LM of a circle, and QR is a chord || LM. Prove that PQ2 + PR2 PL2 + PM2. 74. On the hypotenuse EF of the rt.- ≤ d ▲ DEF a▲ GEF is described outwardly having <GEF = /DEF and GFE a rt.. Prove that GFE : ▲ DEF = GE : ED. 75. Two quadrilaterals whose diagonals intersect at equal <s are to one another in the ratio of the rectangles contained by the diagonals. 76. P is any point in the side LM of a LMN. The st. line MQ, || PN,meets LN produced at Q; and X, Y are points in LM, LQ respectively, such that LX2 LP.LM and LY2 = = 77. EFP, EFQ are circles and PFQ is a st. line. ER is a diameter of circle EFP and ES a diameter of EFQ. Prove Δ EPR : / EQS as the squares on the radii of the circles. 78. If P is the point of intersection of an external common tangent PQR to two circles with the line of centres, prove that PQ: PR as the radii of the circles. Also, if PCDEF is a secant, prove that PC: PE = PD ; PF 79. A point E is taken within a quadrilateral FGHK such that LEFK = ZGFH and LEKF LGHF. GE is joined. Prove A FEG ||| A FHK. = 80. Through a given point within a circle, draw a chord that is divided at the point in a given ratio 81. From P, a point on the circumference of a circle, tangents PE, PF are drawn to an inner concentric circle. GEFH is a chord, and PE meets the circumference at Q. Prove As PGF, PEH, GEQ similar; also show that GQ2: GP2 = GE: GF. 82. L is the vertex of an isosceles ▲ LMN inscribed in a circle, LRS is a st. line which cuts the base in R and meets the circle in S. Prove that SL.RL = LM2. 85. AS and AT, BP and BQ are tangents from two points A and B to a circle. C, D, E, F are the middle points of AS, AT, BP, BQ respectively. Prove that CD, EF, produced if necessary, meet on the right bisector of AB. (Let O be the centre of the circle; L and M the points where OA, OB cut the chords of contact. Prove A, L, M, B concyclic, etc.) 86. If from the middle point of an arc two st. lines be drawn cutting the chord of the arc and the circumference, the four points of intersection are concyclic. 87. If a st. line be divided at two given points, find a third point in the line, such that its distances from the ends of the line may be proportional to its distances from the two given points. 88. Prove geometrically that the arithmetic mean between two given st. lines is greater than the geometric mean between the two st. lines. 89. A square is inscribed in a rt.-angled triangle, one side of the square coinciding with the hypotenuse: prove that the area of the square is equal to the rectangle contained by the extreme segments of the hypotenuse. 90. Any regular polygon inscribed in a circle is the geometric mean between the inscribed and circumscribed regular polygons of half the number of sides. 91. The diagonal and the diagonals of the complements of the parallelograms about the diagonal of a parallelogram are concurrent. 92. Develop the formula for the area of а A, a+b+c and a, b, c are Solution of 92. In ▲ ABC, draw AXI BC, and let AX B = x. Then CX =α - x. Area of A ABC :: ‡ a2 h2 Andah = = = = = = = (a = ah. b2+ c2 2a (a2 4a2 c2 = b2 + c2)2 c2) (2aca2 b2 + c2) (2ac a2 + b2 - c2) (a+b+c) (a - b + c) (a + b−c) (ba + c) 28 (28 -26) (28 2c) (28 - 2a). - b) (sc), a) (sb) (sc). 93. Show from the diagram how the distance between two points, A, B at opposite sides of a pond may be found by measurements on land. 94. Show from the diagram how the breadth of a river may be found by measurements made on one side of it. 95. Given a st. line AB, construct a continuation of it CD, AB and CD being separated by an obstacle. 96. AB, CD are two lines which would meet off the paper. A B E Draw a st. line which would pass through the point of intersection of AB, CD, and bisect the between them. 27 8 213 17 3 Acute angle:-An ▲ which is < a rt. ▲............. Acute-angled triangle :-AA which has three acute Ls..... mon arm..... ...... Altitude of a triangle :-The length of the from any vertex of Arc of a circle:-A part of the circumference. Axis of symmetry :—The line about which a symmetrical figure Centroid :—The point where the medians of a ▲ intersect one 69 Chord of a circle:-The st. line joining two points on the cir- 17 ..... Chord of contact:-The st. line joining the points of contact of 174 17, 141 (The name circle is also used for the area inclosed by the circumference). 17 Circumscribed circle:-A circle which passes through all the ver- 143 Coincide:-Magnitudes which fill exactly the same space are said measure. . . . . . . 4 214 Complementary angles :-Two Ls of which the sum is one rt. 4. 13 be made to fit the other exactly... Consequent :--The second term of a ratio.... Converse propositions :-Two propositions of which the hypothesis 15 213 41 |