Question Bank  Geometry  Hemant Malhotra

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

Q95) In a plane there are 37 straight lines, of which 13 pass through the point A and 11 pass through point B. Besides, no three lines pass through one point, no line passes through both points A and B, and no two are parallel. Fibd the number of points of intersections of the straight lines?
a. 535
b. 525
c. 235
d. 355

Q96) A quadrant of a circle of radius 12 cm is cut out and the remaining part is folded to form a cone. Find the surface area (in sq cm) of the cone.
a) 189 𝜋
b) 81 π
c) 108 π
d) None of these

Q97) The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27 th of the volume of the given cone, at what height above the base is the section made?

Q98) A conical cup is filled with icecream. The ice cream forms a hemispherical shape on its open top. The height of the hemispherical part is 7 cm. The radius of the hemispherical part equals the height of the cone. Then Find the volume of ice cream

Q99) A large sphere is lying in a flat field on a sunny day. At a certain time the shadow of the sphere reaches to a distance of 10 metre from the point where the sphere touches the ground. At the same instant a metre stick held vertically with one end on the ground and at a distance of 8 m from the point where sphere touches the ground casts a shadow of length 2 metre. What is the radius of the sphere (assume the sun rays are parallel and metre stick is a line segment)

Q100) What is the minimum value of sinӨ + cos Ө + secӨ + cosecӨ + tanӨ + cotӨ

2set can be triangle formation

The insect can go from A to B either in exactly 3 steps or in exactly 5 steps.
No: of ways to go from A to B in exactly 3 steps = (3C1) * (2C1) = 6 ways.because at vertex A it has to choose 1 edge out of 3 which it will do in 3C1 ways. and at each vertex 1 or 3 or 5 it has to choose 1 edge out of 2 edges which it will do in 2C1 ways.
No: of ways to go from A to B in exactly 5 steps = No: of ways to go from A to B in exactly 5 steps when no edge is traversed more than once + No: of ways to go from A to B in exactly 5 steps when one of the edge is traversed more than once.
No: of ways to go from A to B in exactly 5 steps when no edge is traversed more than once = 6 ways.
now lets find out the number of ways when insect goes from A  1  A, that is it has completed 2 steps and is back to A from which it has to reach B in remaining 3 steps which it can do in 6 ways.
so when edge A1 is repeated = 6 ways. when edge A3 is repeated = 6 ways, when edge A5 is repeated = 6 ways.
hence (6 + 6 + 6) = 18 ways
now say the insect go from vertex A to any of the three vertex . lets say it goes from A to 1 here when edge 12 is repeated then insect has 2 ways and when 16 is repeated insect has 2 ways. so total 4 ways, hence 3C1*4 = 12 ways.one more case is there when insect travels from A1232B third side is repeated here also 6 ways will be there.
hence total number of ways = 6 + 6 + 18 + 12 + 6 = 48 ways

Let A be the point at which the goat is tied.
Now the radius of the circle is 21
So Total area of the circle will be pie r^2 = pie * 21 *21
But it covered 3/4 th of the circle + Some extra part
So it will be 3/4 (pie * 21^2) + Red part + Yellow part
= 3/4( pie * 21^2) + 1/4(pie * 14^2) + 1/4(pie * 7^2)
= 1232
OA = A[credits : @hemant_malhotra]

@hemant_malhotra 218............

Funda 1 : In any triangle, the orthocenter, circumcenter and centroid are collinear (Euler line)
Funda 2 : On the Euler line the centroid G is between the circumcenter O and the orthocenter H and is twice as far from the orthocenter as it is from the circumcenterC(x, y) ........ G(1, 1) ................. O (5, 3)
CG = 2GOSo point G divides CO in the ratio 1 : 2. Time for the next funda!
Funda 3 : Section Formula
If point P lies on segment AB and satisfies AP : PB = m : n then P = ( (mx2 + nx1)/(m + n), (my2 + ny1)/(m + n) )
Here,
1 = (5 + 2x)/3 => x = 1
1 = (3 + 2y)/3 => y = 3So C is (1, 3)
Now try this.
If the orthocentre of a triangle is (3,5) and the circumcentre is (6,2) then what is the centroid?

@hemant_malhotra wouldn't 57 be the maximum common median ?

@hemant_malhotra Ans 7.4?
r= sqrt(6.75^2+3.05^2)= 7.41 approx.

@hemant_malhotra Ans 26 ?

@hemant_malhotra Sir, 49? Two triangles can be formed, one with ABC 24 area and another with ACD 25 area.