Quant Boosters  Geometry  Raman Sharma  Set 1

Number of Questions: 30
Topic: Geometry
Answer key available?: Yes
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, ADBC, ∠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 incentre. 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/sSolving we get, a = 2^1/2 c
Then using Pythagoras theorem in triangle ABC AD = a/2
So angle BAD =45 degrees
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]