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Try this beautiful problem from PRMO, 2012 based on Triangle.

## Triangle | PRMO | Problem 12

In $\triangle ABC$ we have $AC=BC=7$ and $AB=2$.Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$ .what is the length of the segment $DB$?

• $5$
• $3$
• $7$

### Key Concepts

Geometry

Triangle

Pythagoras

But try the problem first…

Answer:$3$

Source

PRMO-2012, Problem 7

Pre College Mathematics

## Try with Hints

First hint

Given that $AC=BC=7$ & $CD=8$.we have to find out $BD$.Let $BD=x$.we draw a perpendicular from $C$ to $AB$ at the point $M$.Therefore clearly $\triangle CMB$ & $\triangle CMD$ are right angle.so if we can find out the value of $CM$ from the $\triangle CMB$ then we can find out the value $BD$ from the $\triangle CMD$

Can you now finish the problem ……….

Second Step

From the above diagram,In $\triangle CMB$ we can say that $CM=\sqrt{49-1}=4\sqrt 3$

Given $AB=2$ and $M$ is the mid point of $\triangle ABC$ (As AC=BC=7,Isosceles triangle),

Therefore $BM=1$, So $MD=x+1$

Final Step

From the $\triangle CMD$, $(X+1)^2+(4\sqrt 3)^2=64$ $\Rightarrow x=3,-5$

we will take the positive value ,so $BD=3$