## Chittagong-Higher secondary 2016

For students of class 11-12 (age 16+)
mizan_24
Posts: 1
Joined: Mon Dec 24, 2018 12:37 pm

### Chittagong-Higher secondary 2016

The lengths of the sides of the rectangle ABCD are 10 and 11. An equilateral triangle is drawn in such a way that no point is situated outside ABCD. The maximum area of the triangle can be expressed as (p√q)/r where p, q and r are positive integers and q is not divisible by any square number. p+q+r=?

I have determined the maximum area of the triangle (221√3 - 400).But the matter of irony is that the area isn't in the form as mentioned in the question.

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: Chittagong-Higher secondary 2016

mizan_24 wrote:
Mon Dec 24, 2018 12:56 pm
The lengths of the sides of the rectangle $ABCD$ are $10$ and $11$. An equilateral triangle is drawn in such a way that no point is situated outside ABCD. The maximum area of the triangle can be expressed as $\dfrac{p\sqrt q}{r}$ [where $p, q$ and $r$ are positive integers and $q$ is not divisible by any square number. $p+q+r=?$

I have determined the maximum area of the triangle $(221\sqrt 3 - 400)$.But the matter of irony is that the area isn't in the form as mentioned in the question.
What is the question number?

Ragib Farhat Hasan
Posts: 62
Joined: Sun Mar 30, 2014 10:40 pm

### Re: Chittagong-Higher secondary 2016

$(221\sqrt 3 - 400)$ equals to a negative value. So your solution is wrong.

Go back to the drawing boards!

Ragib Farhat Hasan
Posts: 62
Joined: Sun Mar 30, 2014 10:40 pm

### Re: Chittagong-Higher secondary 2016

The solution is as such=>

Let $ABCD$ be a rectangle where $AB$ is $10$ and $BC$ is $11$

Take the base of an isosceles triangle to be $BC$. Let the longer sides be $x$.

So,

$x^2=(5\frac{1}{2})^2 + 100$
$x^2= 130\frac {1}{4}$

Therefore, $x > 11$

Now, it is confirmed that $BC$ can be used as the base of our required equilateral triangle.

Therefore, area of the equilateral triangle=>

$\frac{1}{2} \times 11 \times 11 \times sin (60)= \frac {121\sqrt3}{4}$

Hence, $p= 121, q=3, r=4$ which gives the final answer $p+q+r= 121+3+4= 128.$