Quant Boosters  Geometry  Raman Sharma  Set 2

Q12) The parallel sides of Trapezium is 6 and 8 units respectively. Find the length of the line segment parallel to these sides and which divides the area of trapezium in equal halves.
A. 4 √3
B. 5 √ 2
C. 7
D. 7/√2[OA: Option b]


credits : @swetabh_kumar
let length be x..so 1/2 * (6x) * h1= 1/2 * (1/2 * 14 * h2)..h1 is the height of the smaller trapezium...h2 is overall heght.
so (6+x)h1=7h2
now the line of legth x is divided into length 6 and 2 equal corner segments of legth (x6)/2 each.
by similar triangles, h1/h2= (x6)/2.
plug in the 1st eqn7/(6+x)=(x6)/2
x^2  36 = 14
x = 5 root 2

Q13) What is the ratio of area of a regular dodecagon (a polygon with 12 sides) to the area of a regular hexagon if both the polygons are inscribed in the same circle?
[OA: 2 : √3]

Q14) In a triangle ABC AB = AC and angle BAC = 100°. AB is produced to D such that AD = BC. find angle BCD.
[OA: 10]

Q15) There are 7 smaller squares inside the large rectangle. If the green square has side length 1 and the brown square has side length 2, what is the area of the rectangle?
[OA: 63]

Q16) Let ABCDE be a regular pentagon. Construct an equilateral triangle PAB with point P inside the pentagon. Find the measure of the angle PCE.
[OA: 6]

Q17) QR = ?
[OA: 18]

Hint: Use Appollonius Theorem
PQ^2 + PR^2 = 2 * (x^2+279)
=>2 x^2 = 576 + 144  558
=>x = 9
QR = 2x = 18

Q18) A, B and C are the vertices of a triangle of area 60 cm^2. Let AD be the median drawn from vertex A to side BC and BY be the median from vertex B to AD. If BY is extended to meet AC at E, what is the area of triangle AYE?
[OA : 5 cm^2]

Q19) A circle is drawn such that its diameter coincides with the median BD of an equilateral triangle ABC of side length (4/ rt3) cm as shown in the picture. Find the area of the triangle ABC outside the circle.
OA : (π√3  2π)/6

Q20) The semi perimeter of a right angled triangle is 126cm and shortest median is 53cm.what is the area of the triangle which has the largest median as its longest side?
[OA : 1260 cm^2]


Q21) In a triangle ABC, median AM is drawn such that it divides ∠BAC in the ratio 1:2. AM is extended to D such that ∠ABD = 90. Given AC = 12 cm Find AD.
[OA: 24 cm]


Q22) In Triangle ABC Median from A is perpendicular to Median from B. If AC = 6 and BC = 7, then what is AB?
[OA: root(17)]

Method 1:
Method 2:

Q23) In Triangle ABC, AB = 16 and medians AD and BE are 12 and 18 respectively. Find the area of triangle ABC
[OA: 18 root(55) ]


Q24) XYZ is a right angled triangle, right angled at Z. If the lengths of two of the three medians of the triangle are 4√13 units and 10 units, what is the maximum possible area (in sq.units) of the triangle XYZ?
a) 48
b) 72
c) 96
d) 108[OA: 96]