circles for prmo

By
1)

Solution 


2)                                                       

A circle of radius 1 is tangent to a circle of radius 2. The sides of $\triangle ABC$ are tangent to the circles as shown, and the sides $\overline{AB}$ and $\overline{AC}$ are congruent. What is the area of $\triangle ABC$?

SOLUTION 

Note that $\triangle ADO_1 \sim \triangle AEO_2 \sim \triangle AFC$. Using the first pair of similar triangle, we write the proportion:

$\frac{AO_1}{AO_2} = \frac{DO_1}{EO_2} \Longrightarrow \frac{AO_1}{AO_1 + 3} = \frac{1}{2} \Longrightarrow AO_1 = 3$

By the pythagorean  Theorem we have that $AD = \sqrt{3^2-1^2} = \sqrt{8}$.

Now using $\triangle ADO_1 \sim \triangle AFC$,

$\frac{AD}{AF} = \frac{DO_1}{FC} \Longrightarrow \frac{2\sqrt{2}}{8} = \frac{1}{FC} \Longrightarrow FC = 2\sqrt{2}$

The area of the triangle is $\frac{1}{2}\cdot AF \cdot BC = \frac{1}{2}\cdot AF \cdot (2\cdot CF) = AF \cdot CF = 8\left(2\sqrt{2}\right) = \boxed{16\sqrt{2}\ \mathrm{(D)}}$.

3) A semi-circle of diameter 1 unit sits at the top of a semi-circle of diameter 2 units. The shaded region

inside the smaller semi-circle but outside the larger semi-circle is called a lune. The area of the lune is. (KVPY 2014)


4)The angle bisectors BD and CE of a triangle ABC are divided by the incentre I in the rators 3 : 2 and 2 :1
respectively. Then the ratio in which I divides the angle bisector through A is.
(A) 3 : 1 (B) 11 : 4                         (C) 6 :5 (D) 7 : 4

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We are working hard to increase questions  and will soon add more questions beacause of time management for students we are posting this pre matured post

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