HARD
Earn 100

Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

Important Questions on Theorems regarding Similarity

MEDIUM
D is the midpoint of side BC of ABC. Point E lies on AC such that CE=13AC.BE and AD intersect at G. What is AGGD:
MEDIUM
The area of ABC is 44 cm2. If D is the midpoint of BC and E is the midpoint of AB. Then the area in cm2 of BDE will be:
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MEDIUM
In PQRQ=85° and R=65°. Point S and T are on the sides PQ and PR, respectively such that STR =95°, and the ratio of the QR and ST is 9:5. If PQ=21.6 cm, then the length of PT is:
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EASY
In ABC, AC=8.4 cm and BC=14 cm. P is the point on AB such that CP=11.2 cm and ACP=B. What is the length (in cm) of BP?
HARD

Let ABC be a triangle and M be a point on side AC closer to vertex C than A. Let N be a point on side AB such that MN is parallel to BC and let P be a point  on side BC such that MP is parallel to AB. If the area of the quadrilateral BNMP is equal to 518 of the area of ΔABC, then the ratio AM/MC equals

EASY

In the given figure, if DEBC, AD=2.5 cm, DB=3.5 cm and EC=4.2 cm, then the measure of AC is:

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HARD
In a  ABC, MNBC, the area of quadrilateral MBCN is equal to 130 cm2. If AN:NC is 4:5, then the area of AMN is:
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EASY
If ABC~PQRarABCarPQR=94, AC=12 cm, AB=18 cm and BC=15 cm then PR is equal to _____. 
MEDIUM
Suppose BC is a given line segment in the plane and T is a scalene triangle. The number of points A in the plane such that the triangle with vertices A,B,C (in same order) is similar to triangle T is
MEDIUM
ABC~RQP and AB=4 cm, BC=6 cm and AC=5 cm. If ar(ABC):ar(PQR)=9:4, then PQ is equal to:
MEDIUM
Let ABC~RPQ and ar ABCar RPQ=14. If PQ=4 cm,QR=6 cm and PR=7 cm, then AC is equal to_____.
HARD
In a quadrilateral ABCD, which is not a trapezium, it is known that DAB=ABC=60°. Moreover, CAB=CBD. Then
EASY
The perimeter of similar triangles ABC and PQR are 78 cm and 46.8 cm, respectively. If PQ=11.7, then the length of AB is______.
EASY
In PQR, PQ=24cm and Q=58°S and T are the points on the side PQ and PR, respectively such that STR=122°. If PS=14cm & PT=12cm, then the length of RT is:
MEDIUM
In ABC, D and E are the points on sides AB and BC respectively such that DEAC. If AD:DB=5:3, then what is the ratio of the area of BDE to that of the trapezium ACED?
EASY
In ABC, D is a point on side BC such that ADC=BAC. If CA=12 cm, CB=8 cm then CD is equal to:
HARD
Let ABC be a triangle with sides 51,52,53. Let Ω denote the incircle of ABC. Draw tangents to Ω which are parallel to the sides of ABC. Let r1,r2,r3 be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed r1+r2+r3.
EASY

Consider the following figure shown below and choose which of the following equation is CORRECT about the similarity of both triangles?

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HARD
Let C1,C2 be two circles touching each other externally at the point A and let AB be the diameter of circle C1. Draw a secant BA3 to circle C2 , intersecting circle C1 at a point A1A, and circle C2 at points A2 and A3. If BA1=2,BA2=3 and BA3=4 then the radii of circles C1 and C2 are respectively
HARD

In a triangle ABC with A<B<C, points D,E,F are on the interior of segments BC,CA,AB respectively. Which of the following triangles cannot be similar to the triangle ABC ?