Find the perimeter of the triangle

[anemone]

The two shortest sides of a right-angled triangle, $a$ and $b$ satisfy the inequality \(\displaystyle \sqrt{a^2-6a\sqrt{2}+19}+\sqrt{b^2-4b\sqrt{3}+16}\le3\).

Find the perimeter of this triangle.

 

[kaliprasad]

The two shortest sides of a right-angled triangle, $a$ and $b$ satisfy the inequality \(\displaystyle \sqrt{a^2-6a\sqrt{2}+19}+\sqrt{b^2-4b\sqrt{3}+16}\le3\).

Find the perimeter of this triangle.

Spoiler

=> sqrt((a - 3 sqrt(2))^2 + 1) + sqrt((b - 2 sqrt(3))^2 + 4) < = 3

possible onlly if a = 3 sqrt(2) and b = 2 sqrt(3)

so c(diagonal) = sqrt(30)

so perimeter = 3 sqrt(2) + 2 sqrt(3) + sqrt(30)

- - - Updated - - -

 

[anemone]

Spoiler

=> sqrt((a - 3 sqrt(2))^2 + 1) + sqrt((b - 2 sqrt(3))^2 + 4) < = 3

possible onlly if a = 3 sqrt(2) and b = 2 sqrt(3)

Hi kaliprasad,

Thanks for participating and I think it's necessary to state why the given inequality is true iff $a = 3 \sqrt{2}$ and $b = 2 \sqrt{3}$, what do you think?

Or perhaps it's very obvious and it's just me don't see how it is so?(Tongueout)

 

[kaliprasad]

Hi kaliprasad,

Thanks for participating and I think it's necessary to state why the given inequality is true iff $a = 3 \sqrt{2}$ and $b = 2 \sqrt{3}$, what do you think?

Or perhaps it's very obvious and it's just me don't see how it is so?(Tongueout)

I think I owe an explanation

Spoiler

sqrt((a - 3 sqrt(2))^2 + 1) + sqrt((b - 2 sqrt(3))^2 + 4) < = 3

now we are having

sqrt(1 + x) + sqrt(4 + y) <= 3 with x,y > 0

if x = 0 and y =0 then LHS = 3
if x > 0 then y = 0 then LHS = 2 + sqrt(1+x) > 2 + 1 > 3
similarly for y > 0 and for x and y > 0 LHS > 3

x = (a - 3 sqrt(2))^2
y = (b - 2 sqrt(3))^2

or

Spoiler

lowest value of LHS = 3 when - (a - 3 sqrt(2))= 0 and (b - 2 sqrt(3)) = 0 then only condition is satisfied

 

[CellCoree]

I would first clarify that the given inequality is not enough information to determine the exact values of $a$ and $b$, and therefore the perimeter of the triangle cannot be accurately calculated. However, assuming that the given inequality is true, we can make some general observations about the perimeter of the triangle.

Since the given triangle is a right-angled triangle, we can use the Pythagorean theorem to find the length of the third side, $c$, as $c = \sqrt{a^2 + b^2}$. Therefore, the perimeter of the triangle can be expressed as $P = a + b + \sqrt{a^2 + b^2}$.

Based on the given inequality, we can make some general statements about the possible values of $a$ and $b$. For example, we know that both $a$ and $b$ must be positive in order for the square roots to be real. Additionally, since the square root expressions must be less than or equal to 3, we can set up the following inequalities:

$\sqrt{a^2-6a\sqrt{2}+19} \le 3$ and $\sqrt{b^2-4b\sqrt{3}+16} \le 3$

Solving these inequalities, we get $a \ge 3$ and $b \ge 4$. This means that the perimeter of the triangle must be greater than or equal to $3 + 4 + \sqrt{3^2 + 4^2} = 15$. However, without knowing the exact values of $a$ and $b$, we cannot determine the exact perimeter of the triangle.

In conclusion, as a scientist, I would note that the given inequality provides some constraints on the possible values of $a$ and $b$, but it is not enough information to accurately determine the perimeter of the triangle. Further information or equations would be needed to find the exact perimeter.