# Quant Boosters - Geometry - Raman Sharma - Set 1

• Number of Questions: 30
Topic: Geometry
Source: Selected Questions from CAT prep forums

• Q1) In a triangle ABC , AC = 2BC , Angle C = 90° and D is the foot of perpendicular from C onto AB . A circle with diameter AD intersects the segment AC at E. Find AE/EC .

[OA: 4]

• Q2) [OA: 60]

• Q3) In a trapezium ABCD, AD||BC, ∠DAB and ∠ADC are acute angles. If AB and DC are extended to intersect at E and then perpendicular from E is drawn to AD. The perpendicular gets bisected by BC. If AB = 8 cm and AD = 10 cm, then what is the value of 2AC² + BD²?

[OA: 342]

• Method 1: • Method 2 : • Q4) Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find angle BAC

[OA: 90 degrees]

• [credits : Bulbuul Dev]
Let angle bisector of angle BAC intersects BC at D. Since ABC is isosceles it will also be the altitude of the triangle.
Now,
BC = AB + AI
a = c + AD - r
a = c + (c^2 - (a/2)^2)^1/2 - Area of triangle/s

Solving we get, a = 2^1/2 c
Then using Pythagoras theorem in triangle ABC AD = a/2
and angle bac = 90 degrees

• Q5) The shorter leg of a right triangle with integral side is 36 units . Find the difference between the largest and smallest possible areas of this triangle.

[OA: 4950]

• Q6) Chords AB and CD of a circle intersect at E and are perpendicular to each other. Segment AE, EB and ED are of length 2 cm, 6 cm and 3 cm respectively. Then the length of the diameter of the triangle is

[OA: √65]

• Method 1 : • Method 2: • Shortcut : R ^2 = ( AE^2 + BE^2 + CE^2 + DE^2 )/4

• Q7) 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

[OA: e]

• when parallel line is between BC and A, it will be 9:4
when BC is between parallel line and A, it will be 1:4
So, E .

• Q8) In a triangle ABC , AB = 28 , BC = 21 and CA = 14. Points D and E are on AB with AD = 7 and angle ACD = angle BCE . Find the length of BE.

[OA: 12]

• Q9) In the figure below AM : MD = CD : DB = 3 : 2. Find AE : EC [OA : 3 : 5]

• • Area - Base ratio : Ratio of area of triangles having base on the same line and third vertex common is equal to the ratio of the length of the bases of the triangles.

• Q10) Two altitudes of a triangle are of 12 and 20 units repectively. Find the sum of all possible integral lengths of the third altitude.

[OA: 407]

62

47

86

58

63

43

102

61