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Calculate the length of LM in the isosceles right triangle ∆ KLM

- safia21

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Calculate the length of LM in the isosceles right triangle ∆ KLM

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- safia21

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- heisenberg

You can use the pythagorean theorem since it has a right angle. Are you familiar with it?

- safia21

no im kinda confused

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## More answers

- heisenberg

do you know what a hypotenuse is?

- safia21

yes

- heisenberg

so the pythagorean theorem is:
\[ a^2 + b^2 = c^2 \]
where c is the hypotenuse. since this is an isosceles triangle, a = b, don't you think?

- safia21

yes

- heisenberg

so can you take it form here?
since a = b, \( a^2 + a^2 = c^2 \)

- safia21

what is a and what is like what do you plug in

- heisenberg

well let me ask, what is a hypotenuse?

- safia21

the longset side of a right triangle

- heisenberg

very good! which is the side *directly* across from the right angle.
so the other two sides would be 'a' and 'b', but this triangle is isosceles so a = b. therefore a is an unknown that we want to solve for and 'c' is the length of the hypotenuse

- heisenberg

so if we have \(a^2 + a^2 = c^2\) where \( c = 36 \)
we only have 1 unknown so we should be able to solve this like a regular algebra problem.

- safia21

ok

- heisenberg

are you still confused? all that's left is to simplify this equation and solve.

- safia21

okay i got 36?

- heisenberg

show me your steps. i don't that's right.

- heisenberg

think*

- heisenberg

what is \( a^2 + a^2 \) ?

- heisenberg

(@victor, please consider that the best way for a person to learn something is to have the person discover it on their own rather than just giving that person an answer.)

- safia21

yes i need the steps!

- heisenberg

there's no reason you can't do it yourself, though. you have an equation. where are you getting stuck? show me your work and i can help you.

- safia21

can we start from the beginning im soo sorry!

- heisenberg

the pythagorean theorem states that (for a right triangle):
\[ a^2 + b^2 = c^2\]where c is the hypotenuse.
we have the hypotenuse in this case, as you pointed out, and it equals 36, agree?

- anonymous

It's an isosceles triangle so it has two equal sides and therefor two equal angles. If you add the inner angles of any triangle the result will be 180º. You already know 1 angle = 90º. 180-90=90 And since the two other angles are equal you have 90/2=45. Now that you have all the angles you can use trigonometry and achieve the value of the sides. I recommend you use "sin" or "tan" (as expressed on a calculator).

- safia21

yes

- heisenberg

but this is an isosceles triangle, so the two remaining sides are equal in length, would you agree?

- safia21

yes

- anonymous

Of course, you only need the value of one side.

- heisenberg

so since the two remaining sides are \(a\) and \(b\), we know that \(a =b \)
therefore we can simplify the equation to this: \( a^2 + a^2 = c^2 \)

- safia21

okay and c2 is 36 right so a2+a2= 36^2

- heisenberg

exactly! so what does \( a^2 + a^2\) equal? we can reduce this to one term.

- safia21

a^3

- heisenberg

not quite. how about this, what does \( x + x \) equal?

- heisenberg

what if i wrote it this way: \( 1x+ 1x \)

- safia21

2x^2?

- heisenberg

that wouldn't work. think about if x = 2,
1x + 1x = 1(2) + 1(2) = 4, but
2x^2 = 2(2)^2 = 8
so those two expressions are not equal.
when you add terms of the same variable, you just add their "coefficients," the number in front of the variable.

- safia21

o okay

- heisenberg

so what is x + x?

- safia21

4

- heisenberg

no x is a variable, not a number. x can be *any* number

- safia21

okay x^2

- heisenberg

no just add the numbers *in front* of the variable.
x = 1x

- safia21

2x

- heisenberg

right! so now let's look at the original equation, what is \( a^2 + a^2 \)

- heisenberg

keep in mind that \(a^2 \) is the same thing as \(1a^2\)

- safia21

2x^4

- heisenberg

there are no x's in this equation. that was just as an aside example. the question here is to simplify \(a^2 + a^2\)
you know that \( x + x = 2x\), so use that same *idea* and apply it to the other equation.

- safia21

2a^4

- heisenberg

you don't add the exponents, just the coefficients.
\( x = 1x = x^1 = 1x^1 \) are all the same same

- safia21

2a^2

- safia21

isnt it a2=B2

- heisenberg

very good! so let's look at our original equation:
\(a^2 + a^2 = 36^2 \) =
\(2a^2 = 36^2\)

- heisenberg

yes it is, that's how we eliminated the b^2 and replaced it with another a^2

- safia21

ok

- heisenberg

so now our equation is \(2a^2 = 36^2\)
do you know how to solve algebra problems? that's all this is.

- safia21

so i multiply 36 times 36 and divide it by 2

- heisenberg

absolutely!

- heisenberg

that would be represented like so:
\(a^2 = \frac{36^2}{2} \)
then you just take the square root of both sides to get 'a' = something

- heisenberg

a^2 = 36^2 / 2

- safia21

36 right

- heisenberg

not quite, remember, it's: 36 * 36 / 2

- safia21

ok

- heisenberg

first do 36 * 36, then divide that answer by 2

- safia21

648

- heisenberg

very good!
so we're left with:
\( a^2 = 648\)
just take the square root of both sides and you have your answer!

- safia21

i got 26

- safia21

but thats not one of the anwsers

- heisenberg

what are the answers?

- safia21

a) 18
b) 18√2
c) 36
d) 36√2

- heisenberg

well one of those answers roughly equals the number you got.

- safia21

b

- heisenberg

that's right! congrats :)

- safia21

thanks soo much!

- heisenberg

no problem :) i'm glad you wanted to learn rather than just want the answer. it'll always work out better that way. trust me ;)

- safia21

haha thanks! :)

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