Night Classes Geometry - 2

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Sheena Greene
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1 Geometry - 2

2 Properties of four centres in a triangle Median: Area of ABD = area of ADC

3 Angle Bisector: Properties of four centres in a triangle

4 Angle Bisector: Properties of four centres in a triangle

5 Perpendicular Bisector: Properties of four centres in a triangle

6 Altitude: Properties of four centres in a triangle

7 Properties of four centres in a triangle

8 The sides of ABC whose angles measure 50, 70 and 60 respectively are produced both ways. The exterior angles so formed are bisected to intersect at three points A, B and C. A, B and C are joined together to form a new triangle. The measure of angles of the new triangle are (1) 50, 60 and 70 (2) 55, 55 and 70 (3) 80, 60 and 40 (4) 55, 60 and 65 (5) None of these Question 1

9 Unequal side of an isosceles triangle is 2 cm long. The medians drawn to the equal sides are perpendicular to each other. Find the area of the triangle. (1) 3 cm 2 (2) (3) (4) (5) 2 10 cm cm cm cm Question 2

10 ABC is an isosceles right angled triangle with B = 90. Bisectors of external angles ACY and CAX meet at O. If BC = x, then find the perpendicular distance of O from BY. x (1) 1 (2) x (3) (4) x (5) x x Question 3

11 In ABC, D and E are points on side AC such that AD : DE : EC is 2 : 1 : 1. BD and BE are joined. Point F and G are on BD and BE respectively such that BF : FD = 1 : 2 and BG : GE = 2 : 1. Find the ratio of the area of ( AFG BFG) to the area of the ABC. (1) 2 : 5 (2) 3 : 10 (3) 1 : 3 (4) 1 : 4 (5) 2 : 7 Question 4

12 In a ABC D is the mid point of the side BC and P is a point on AB such that DP is the angular bisector of the ADB If AP : PB = 1 : 3 and AB : AC = 2 : 3, then find the ratio AB : BC. (1) 2 : 3 (2) 1: 2 (3) 2 7 : 5 11 (4) 3 : 7 5 (5) 2 5 : 3 13 Question 5

13 In a triangle PQR, PQ = 12 cm and QR = 4 3 cm. If the measure of PRQ 60, then what is the ratio of the inradius to the circumradius of the triangle PQR? 1 (1) (2) (3) (4) Question 6

14 In ABC, the internal bisector of A meets BC at D. If AB = 4, AC = 3 and A = 60º, then the length of AD is 12 3 (1) 2 3 (2) (3) (4) CAT 2002 Question 7

15 Polygons

16 Polygons

17 Polygons Question 8

18 If PQRSTUVW is a regular octagon, then what is the measure of (1) 135 (2) (3) 115 (4) (5) Polygons ( WPQ PVQ)? Question 9

19 Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90 or concave if the internal angle is 270. If the number of convex corners in such a polygon is 25, the number of concave corners must be (1) 20 (2) 0 (3) 21 (4) 22 (5) 18 Question 10

20 Answer the questions on the basis of the information given below. Three identical regular hexagons with side 2 3 units are drawn on a plane side-by-side as shown in the following figure. Y X Find the minimum possible distance between the points X and Y. (1) 7 5 units (2) 9 3 units (3) 12 2 units (4) 6 7 units (5) 10 2 units Question 11

21 Quadrilaterals

22 Quadrilaterals True of False: 1.If diagonals of a quadrilateral bisect each other, it must be a parallelogram. 2.If diagonals of a quadrilateral are equal, it must be a rectangle. 3.If diagonals of a quadrilateral are perpendicular to each other, it must be a rhombus. 4.If diagonals of a quadrilateral are equal as well as perpendicular to each other, it must be a square. Question 12

23 Quadrilaterals If P, Q, R and S are mid points of sides of then PQRS will be a parallelogram. ABCD,

24 Quadrilaterals True of False: If quadrilateral formed by joining the mid points of sides of a quadrilateral is a rectangle, then the original quadrilateral must be a rhombus. Question 13

25 Quadrilaterals True of False: If quadrilateral formed by joining the mid points of sides of a quadrilateral is a rhombus, then the original quadrilateral must be a rectangle. Question 14

26 Quadrilaterals True of False: If quadrilateral formed by joining the mid points of sides of a quadrilateral is a square, then the original quadrilateral must be a square Question 15

27 ABCD is a square and PBC is an equilateral triangle. If PQ DC, then find the measure of the angle DPQ. A D P Q B C (1) 30 (2) 15 (3) 20 (4) 25 (5) None of these Question 16

28 ABCD is a parallelogram. P is a point on AB such that AP : PB = 3 : 2. Q is a point on CD such that CQ : QD = 7 : 3. If PQ intersects AC at R, then find the ratio AR : AC. (1) 5 : 11 (2) 1 : 2 (3) 4 : 7 (4) 2 : 5 (5) 6 : 13 Question 17

29 PQRS is a parallelogram with PQ = 21 cm, QR = 13 cm and QS = 14 cm. Find the length of other diagonal PR. (1) 28 cm (2) 26 cm (3) 32 cm (4) 36 cm (5) Such a parallelogram is not possible Question 18

30 There are two squares S 1 and S 2 with areas 8 and 9 units, respectively. S 1 is inscribed within S 2, with one corner of S 1 on each side of S 2. The corners of the smaller square divides the sides of the bigger square into two segments, one of length a and the other of length b, where b > a. A possible value of b/a, is: (1) 5 and < 8 (2) 8 and < 11 (3) 11 and < 14 (4) 14 and < 17 (5) > 17 XAT 2014 Question 19

31 ABCD is a rectangle. X and Y are two points on the sides AB and BC respectively. If areas of DAX, YCD and XBY are 5, 4 and 3 square units respectively; what is the area (in square units) of the rectangle ABCD? (1) 20 (2) 4 (3) 8 (4) 16 Question 20

32 In the figure given below, ABCD is a square. It is given that BP : PM = 4 : 1 and DP : PN = 3 : 2. Find the ratio of lengths of CN to CM. A D M P B C (1) 3 : 4 (2) 3 : 5 (3) 4 : 5 (4) 2 : 3 N Question 21

3. Understanding Quadrilaterals

3. Understanding Quadrilaterals Q 1 Name the regular polygon with 8 sides. Mark (1) Q 2 Find the number of diagonals in the figure given below. Mark (1) Q 3 Find x in the following figure. Mark (1) Q 4