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Try this beautiful Geometry problem from PRMO, 2018 based on Medians.

## Medians | PRMO | Problem-13

In a triangle ABC, right-angled at A, the altitude through A and the internal bisector of $\angle A$ have lengths $3$ and $4$, respectively. Find the length of the median through $A$.

• $20$
• $24$
• $13$

### Key Concepts

Geometry

Medians

Triangle

But try the problem first…

Answer:$24$

Source

PRMO-2018, Problem 13

Pre College Mathematics

## Try with Hints

First hint

Now in the Right angle Triangle ABC ,$\angle A=90$.$AD=3$ is Altitude and $AE=4$ is the internal Bisector and $AF$ is the median.Now we have to find out the length of $AF$

Now $\angle CAE = 45^{\circ }= \angle BAE$.

Let$BC = a$, $CA = b$, $AB = c$
so $\frac{bc}{2}=\frac{3a}{2}$ $\Rightarrow bc=3a$

Can you now finish the problem ……….

Second Hint

Therefore,

$\frac{2bc}{b+c} cos\frac{A}{2}=4$

$\Rightarrow \frac{6a}{b+c} .\frac{1}{\sqrt 2}=4$

$\Rightarrow 2\sqrt 2 (b+c)=3a$

$\Rightarrow 8(b^2 + c^2 + 2bc) = 9a^2$

$\Rightarrow 8(a^2 + 6a) = 9a^2$

$\Rightarrow 48a=a^2$

$\Rightarrow a=48$

Can you finish the problem……..

Final Step

Now $AF$ is the median , $AF=\frac{a}{2}=24$