Quant Boosters  Geometry  Raman Sharma  Set 2

Q6) On a circle 26 equidistant points are marked. These points are joined to form triangles. Of the triangles formed, how many of them will have their circumcenter on one of their sides?
[OA : 312]

Q7) ABCD is a square with side length r units .
AB is trisected by two points R and S ,
CD is trisected by two points P and Q ,
AP and DR intersect at E
AP and DS intersect at N
AQ and DR intersect at M
AQ and DS intersect at F
What is the area of the quadrilateral EMFN ?[OA: (r/6)^2]

Credits : Turan Bey

Q8) In the square ABCD , a line through B intersects the extension of CD at E , the side AD at F and the diagonal AC at G if BG = 9 and GF = 3 , then what is the length of EF ?
[OA: 24]

Q9) Two of the sides of a triangle are in the ratio 3:4. The medians to these sides are perpendicular to each other. If the third side of the triangle is 12root(5), find the smaller of the first two sides of the triangle.
[OA: 36]


Q10) The shortest median of a rightangled triangle is 25 units. If the area of the triangle is 336 sq.units, what is the length (in units) of the longest median of the triangle?
[OA: 48.5]


Q11) In triangle ABC, two lines DE and FG are drawn parallel to BC such that they divide AC in the ratio 2:3:5. Find the ratio of area of triangle ABC to area of the trapezium DEGF.
[OA: 100 : 21]

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