Question Bank  Geometry  Hemant Malhotra

Q54) If the area and perimeter of a triangle have same numerical value which is 30 and the longest side of triangle is 13 unit then what is the different between longest side and smallest side of this triangle.

a + b = 17
15 * 2 * (15  a)(a  2) = 900 (using herons formula)
(15  a)(a  2) = 30
a = 5 or 12
So, 5, 12, 13

Q55) An insect starts at vertex A of a certain cube and is trying to reach at vertex B, which is opposite A, in 5 or fewer steps where a step consists travelling along an edge from one vertex to another. The insect will stop as soon as it reaches B. The number of ways in which the insect can achieve its objective is

Q56) In an isosceles right angled triangle,the perimeter is 20m. Find its area

Q57) Find the number of triangles with integer sides if perimeter is 50

Q58) How many triangles with integer sides can be formed with perimeter 19units, having one side odd and two sides even ?

Q59) Triangle ABC is a scalene acute angled triangle. A line passing through the incenter of the triangle divides the triangle into two equal areas. If S is the perimeter of Δ ABC, then CD + CE =?
a. S/2
b. S/3
c. S/4
d. 2S/3
e. 3S/4

Method1  1/2 * r * CD + 1/2 * r * EC =Area of Triangle CED
r/2 [ CD + EC ] = Area of triangle CED
r/2 [ CD + EC ] = 1/2 * r * ( Semi perimeter )
CD + EC = Semi perimeter
CD + EC = S/2 , where S is perimeterMethod2  draw perpendicular from I to EC and CD area of triangle
A = r * s
A(IDC) + Area(IEC) = 1/2 * (total area)
1/2((CD+EC) * r = r * s/2
so CD + EC = s/2

Q60) 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

Q61) A rectangle has area A cm^2 and perimeter P cm , where A and P are positive integers. Which of the following numbers cannot equal A+P ?
A. 100
B. 102
C. 104
D. 106
E. 108

Q62) What is the least perimeter of an obtuseangled triangle with integer sides, whose one acute angle is twice the other?

Q63) The sides a,b & c of a triangle ABC are in GP whose common ratio is 2/3 and the circumradius is 6sqrt(7/209). Find the longest side of the triangle.

Q64) If there are 7 distinct points on a plane with no three of which are collinear, how many different polygons can be formed

Q65) In a triangle ABC, an altitude is drawn from C which meets AB at D. If AB=8 and CD=5, then find the distance between the midpoints of AD and BC.

Q66) ABCD is a square of side 8 cm, it is folded in such a way that point B meets mid point of AD. Find the length of crease

Q67) Find the minimum number of lines required to divide a plane in 79 distinct parts.

One line divide plane in 2 parts
Now 2nd line in 4 parts
So new parts will be 4  2 = 2
Now 3 lines will divide in 6 parts so 6  3 = 3 increased parts
Same in 2N points = N partsSo 2 + 2 + 3 .. N
1 + 1 + 2 ... N
1 + (N) * (N + 1)/2

R = abc / 4A
let side =x,2x/3 and 4x/9 so S=(a+b+c)/2=19x/18
S = 19x/18
A = (x/18)^2 * sqrt(7 * 209)
R = (8a^3/27) /(4 * (x/18)^2 * sqrt(7 * 209)]
R = 24x / sqrt (7 * 209)
now compare
R = 6sqrt(7/209) t
24x / sqrt (7*209) = 6sqrt(7/209)
24x = 42
x =7/4

Q68) In the figure given below, ABCD is a rectangle and GB = DH = 2HC. What is the area (in sq cm)of the shaded region if AD = 1 cm and AB = 3 cm?

Q69) A triangle has vertex A at (0, 3), vertex B at (4, 0), and a vertex C at (x, 5) for some x where 0 < x < 4. If the area of the triangle is 8, what is the value of x?