Question Bank  Geometry  Hemant Malhotra

Q50) In the figure, chord ED is parallel to the diameter AC of the circle. If < CBE = 65 then find tha value of < DEC

Q51) ABC is a triangle where AB = 4, the median AD = 1, then find the minimum value of the angle BAC?
(1) 90°
(2) 120°
(3) 150°
(4) 135°

AB^2 + AC^2 = 2 * (AD^2+BD^2)
AB^2+AC^2=2*(AD^2+((BC/2))^2
4 * AD^2=2b^2+2c^2a^2
4 * AD^2=b^2+c^2+2bc * cosA
so 4=b^2+16+2 * b * 4 * cosA
so cosA=((12b^2)/8b
so min value will be at b=2 * sqrt 3
so cosA=sqrt3/2 so B=150

Q52) Mandakini draws a square ABCD of side 1 unit. She then draws 10 straight lines connecting A to each of 11 equally spaced points lying internally on CD (including C and D). What is the total area (in unit square) of all the possible triangles that can be formed

Q53) Four villages lie at the vertices of a squares of side 1km. What is the smallest length of road needed to link all village together

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