a great challenge to solve.

The perimeter of a right triangle is 24 cm. Three times the length of the longer side minus two times the length of the shorter side exceeds hypotenuse by 2 cm.what are the lengths of all three sides??state method too.

#Algebra

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

$\triangle ABC$ , let it be right angled at $B$ . (Hypotenuse $AC$ ).

Let $AB\leq BC$ .

From $1^{st}$ statement, $AB+BC+CA=24$

From $2^{nd}$ statement, $3\times BC -2\times AB = AC+2$

By Pythagoras' theorem $AC^2=AB^2+BC^2$ .

Now it's simply 3 equations and 3 variables, solving simultaneously gives

$AC=10 , BC=8, AB=6$

Great Challenge (?).... I'm a moderator, can edit titles of only problems, can't edit the titles of Notes...

Aditya Raut - 6 years, 8 months ago

Can u please write the whole calculation so I can understand because up to here I also did easily

naitik sanghavi - 6 years, 8 months ago

Whenever you have $n$ equations and $n$ variables, just go on substituting from one to other and finally you reach 1 variable which leads to all others.

Our current Example is,

(i) $a+b+c=24$ ; (ii) $3a-2b=c+2$ ; (iii) $a^2+b^2=c^2$ .

Use $c=24-(a+b)$ . So now you only have 2 equations and 2 variables.

(i) $3a-2b=24-(a+b)+2$ ; (ii) $a^2+b^2 = (24-a-b)^2$

i.e. $4a-b=26$ , $576+2ab-48a-48b=0$ i.e. $24a+24b-ab=288$

Now do substitution $a = \dfrac{26+b}{4}$

$24\biggl(\dfrac{26+b}{4}\biggr) +24b -\biggl(\dfrac{26+b}{4}\biggr) b = 288\\ \therefore 156+30b-\dfrac{b^2+26b}{4} =288$

This is a quadratic in $b$ , and has positive root $6$ .

Put $b$ value in $a=\dfrac{26+b}{4} = \dfrac{26+6}{4} = 8$

And finally, $c=10$

Sides $6,8,10$ .

Aditya Raut - 6 years, 8 months ago

As fast as possiblePlease help me

naitik sanghavi - 6 years, 8 months ago

Please answer writing each step to the end..plse help me!!!

naitik sanghavi - 6 years, 8 months ago

If someone understood the answer then please answer me writing it step by step upto the end in 12hrs plssssss.its urgent

naitik sanghavi - 6 years, 8 months ago

Done . Next time, give a reasonable heading to your note.

Aditya Raut - 6 years, 8 months ago

Thanks very much...:-)

naitik sanghavi - 6 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$