Question Bank  Geometry  Hemant Malhotra

Q76) Two altitudes of triangle are of 12 and 20 units respectively. Find the sum of all possible integral lengths of third altitude

if a,b,c are three altitudes
then (a * c)/(a+c) < b < (ac/(ac)
so (12 * 20)/(32) < b < 240/(2012)
(240/32) < b < 240/8
30/4 < b < 30
so 7.5 < b < 30
0

Q77) A circle is drawn in a Cartesian plane such that there is no point (x, y), where x and y are integers, that lie inside the circle. What is the maximum possible area of such circle

Q78) The radius of the circle with centre O is under root 50cm. A and C are two points on the circle and B is a point inside the circle. the length of AB is 6cm and BC is 2cm. The angle ABC is a right angle. Find the square of the distance OB.

Q79) A circle C1, of radius four units, touches another circle C2, of radius nine units, and the line ℓ1 is a direct common tangent to both the circles. Which of the following would be the radius of another circle C3, which touches both circles C1 and C2 and also the line ℓ1?
a) 1.32 units
b) 1.44 units
c) 1.58 units
d) 1.76 units

Q80) What is the maximum number of chords of length R that can be drawn in a circle of radius R, such that no two chords intersect inside the circle?

Q81) On a circle there are 10 points each of which is connected with each other with a straight line. How many triangles will be formed which lies completely inside the circle?

Q82) The inscribed circle of an isoscles triangle ABC is tangent to side AB at point D and bisects the segment CD. If CD = 6√2, which among the following can not be true about ABC?
(a) The perimeter is 24
(b) It's obtuse angled
(c) The bisector segment of the smallest angle is 6√2
(d) The perimeter is 28
(e) none

Q83) Given inside a circle whose radius is 13cm , is a point M at a distance 5cm from the centre of circle. A chord AB 25cm is drawn through M the length of the segment into which the chord AB is divided by the point M in cm are :
a. 12,13
b. 14,11
c. 15,10
d. 16,9

Q84) From a point outside a circle of radius 15 cm, two tangents are drawn to the circle. The angle between the tangents is 74°. Find the area of the triangle formed by the tangents and the chord joining the points of contact. (sin37° = 0.6)

Q85) Given a triangle whose sides are 24, 30 and 36 cm. Find the radius of the circle which is tangent to the shortest and longest side of the triangle and whose center lies on the third side.

Q86) An ant starts from point A and crawls along the surface of the cylinder to reach the point B, vertically above A. The path followed by the ant is equal to that of 4 identical spirals. Find the radius of the circle that circumscribes a square such that the perimeter of the square is equal to the distance traversed by the ant. The diameter of the cylinder is 3/π units and the height is 16 units.

Q87) ABC is a triangle and P is a point on a line parallel to BC such that ratio of distance of A from the line and distance of BC from the line is 5:4, then what will be the ratio of area of triangle ABC and triangle PBC
a) 4 : 9
b) 1 : 4
c) 1 : 5
d) 9 : 4
e) can not be determined

Q88) Rahul made a right triangle with a certain number of matches. Then he use all those matches to make a different shaped right triangle (that is to have at least one different side from the first one). What is the smallest number of matches that Rahul can use to do so

Q89) A triangle has its longest side as 38 cm. If one of the other two sides is 10 cm and the area of the triangle is 152 sq cm, find the length of the third side.

Q90) In triangle ABC, D is the mid point of side BC. It is given angle DAB = angle BCA and angle DAC = 15 degree. If O is the circumcentre of ADC then find the measure of angles of triangle AOD

Q91) A triangle is called a ptriangle if the length of each of its side (in units) and its area (in sq. units) are integers. How many of the triangles with the sides (in units) given below are ptriangles?
(i) 4, 5 and 6
(ii) 3, 4 and 5
(iii) 5, 8 and 9
(iv) 5, 6 and 8(a) One
(b) Two
(c) Three
(d) Four

Q92) How many set of data , given below, is (are) sufficient to construct an unique triangle ABC. [P= Perimeter, A , B, & C are angles & a,b & c are sides to corresponding angles]
I. P = 16 cm , c = 9 cm, A= 60°
II. a=b= 12 cm, A= 60°
III. A= 60° ; B = 30° ; C = 90°
IV. A= 60° ; B = 45° ; a = 4 cm, b= 3 cma. Only one set of data
b. Two sets of data
c. Three sets of data
d. By all we can construct unique triangle
e. None of these

Q93) If 2x + y  6 = 0, x  y + 3 = 0 and 2y + 1 = 0 form a triangle, then how many points in the interior of the triangle have integer coordinates ?
a) 12
b) 13
c) 14
d) 11

Q94) Five points A,B,C,D,E lie on a line L1 and points P, Q, R, S and T lie on a line L2. Each of the five points on L1 is connected to each of the points on L2, by means of straight lines terminated by the points. Then Excluding the given points, the maximum number of points at which the lines can intersect is