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
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You can use the pythagorean theorem since it has a right angle. Are you familiar with it?
safia21
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no im kinda confused
heisenberg
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do you know what a hypotenuse is?
safia21
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yes
heisenberg
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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
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yes
heisenberg
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so can you take it form here?
since a = b, \( a^2 + a^2 = c^2 \)
safia21
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what is a and what is like what do you plug in
heisenberg
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well let me ask, what is a hypotenuse?
safia21
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the longset side of a right triangle
heisenberg
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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
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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
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ok
heisenberg
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are you still confused? all that's left is to simplify this equation and solve.
safia21
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okay i got 36?
heisenberg
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show me your steps. i don't that's right.
heisenberg
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think*
heisenberg
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what is \( a^2 + a^2 \) ?
heisenberg
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(@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
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yes i need the steps!
heisenberg
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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
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can we start from the beginning im soo sorry!
heisenberg
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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?
Victor_Hugo
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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
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yes
heisenberg
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but this is an isosceles triangle, so the two remaining sides are equal in length, would you agree?
safia21
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yes
Victor_Hugo
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Of course, you only need the value of one side.
heisenberg
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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
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okay and c2 is 36 right so a2+a2= 36^2
heisenberg
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exactly! so what does \( a^2 + a^2\) equal? we can reduce this to one term.
safia21
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a^3
heisenberg
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not quite. how about this, what does \( x + x \) equal?
heisenberg
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what if i wrote it this way: \( 1x+ 1x \)
safia21
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2x^2?
heisenberg
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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
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o okay
heisenberg
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so what is x + x?
safia21
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4
heisenberg
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no x is a variable, not a number. x can be *any* number
safia21
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okay x^2
heisenberg
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no just add the numbers *in front* of the variable.
x = 1x
safia21
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2x
heisenberg
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right! so now let's look at the original equation, what is \( a^2 + a^2 \)
heisenberg
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keep in mind that \(a^2 \) is the same thing as \(1a^2\)
safia21
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2x^4
heisenberg
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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
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2a^4
heisenberg
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you don't add the exponents, just the coefficients.
\( x = 1x = x^1 = 1x^1 \) are all the same same
safia21
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2a^2
safia21
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isnt it a2=B2
heisenberg
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very good! so let's look at our original equation:
\(a^2 + a^2 = 36^2 \) =
\(2a^2 = 36^2\)
heisenberg
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yes it is, that's how we eliminated the b^2 and replaced it with another a^2
safia21
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ok
heisenberg
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so now our equation is \(2a^2 = 36^2\)
do you know how to solve algebra problems? that's all this is.
safia21
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so i multiply 36 times 36 and divide it by 2
heisenberg
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absolutely!
heisenberg
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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
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a^2 = 36^2 / 2
safia21
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36 right
heisenberg
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not quite, remember, it's: 36 * 36 / 2
safia21
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ok
heisenberg
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first do 36 * 36, then divide that answer by 2
safia21
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648
heisenberg
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very good!
so we're left with:
\( a^2 = 648\)
just take the square root of both sides and you have your answer!
safia21
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i got 26
safia21
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but thats not one of the anwsers
heisenberg
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what are the answers?
safia21
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a) 18
b) 18√2
c) 36
d) 36√2
heisenberg
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well one of those answers roughly equals the number you got.
safia21
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b
heisenberg
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that's right! congrats :)
safia21
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thanks soo much!
heisenberg
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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
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haha thanks! :)