I am having trouble with this question.
The perimeter of an isoceles right triangle is 16 + 16*sq. root of 2. What is the length of the hypotenuse?
I know that the ratio is 1:1:sq. root 2, but how do you determine the length of individual sides when given the entire perimeter?
Thanks.
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- rjdog21
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- joey.yiz
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Re: Perimeter of right triangle
16+16*sqrt(2) = x + x + sqrt(2)*x, hence
8*sqrt(2) * (2+sqrt(2)) = x * (2+sqrt(2)), hence
x = 8*sqrt(2), and hence
the length of the hypotenuse = x * sqrt(2) = 16
8*sqrt(2) * (2+sqrt(2)) = x * (2+sqrt(2)), hence
x = 8*sqrt(2), and hence
the length of the hypotenuse = x * sqrt(2) = 16
- RonPurewal
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Re: Perimeter of right triangle
i'm in a good mood today, so i'm going to answer this question despite the fact that you've violated the forum rules, which say:
POST ALL QUESTIONS IN FULL, INCLUDING ANSWER CHOICES.
in fact, your omission of the answer choices shows that your priorities are all wrong. there are lots of problems on which you can "plug in" - whether plugging in your own numbers or the numbers that actually appear in the answer choices - and use the answer choices to figure out the solution, even if you have no idea how to solve the problem in the "conventional" way.
this is one of those problems, as you'll see below.
if you'd even think of posting a problem without its answer choices, that tells me that you have no idea that such alternate methods exist. if that's the case, then you're needlessly sacrificing a fair # of points from the math section, as such methods are often the easiest way to go.
first of all, since this is a multiple-choice question, you don't have to.
you can just plug in an answer choice and work backwards.
since you haven't provided the answer choices, i can only start with the correct answer.
there should be a choice that says 16.
plug this in for the hypotenuse.
in that case, the length of each leg is 16 divided by √2, which is 8√2.
therefore, the perimeter is 2(8√2) + 16, or 16√2 + 16.
this is the correct perimeter, so you're good to go.
done.
(by the way, you should always start with choice (c) when you use this method of substituting the answer choices into the problem. that way, if it's too small, you can eliminate it along with the two smaller choices; if it's too big, you can eliminate it along with the two larger choices.)
--
another thing that can be helpful here is to realize that you can write the ratio not only as 1:1:√2, but also as √2:√2:2,
the perimeter of the former is 2 + 1√2. unfortunately, the given perimeter (16√2) isn't a multiple of this.
the perimeter of the latter is 2 + 2√2. your perimeter is exactly 8 times this value. therefore, your triangle is 8 times the base ratio of √2:√2:2, or 8√2:8√2:16.
--
finally, you can use algebra. the "base" triangle has perimeter 2 + √2, so you can find the desired multiple of this base triangle by dividing your perimeter, 16 + 16√2, by the perimeter of the base triangle, 2 + √2.
that quotient is (16 + 16√2) / (2 + √2) = 8√2. (you have to use the conjugate method to get this.)
therefore, you have 8√2 times the basic 1:1:√2 template, so your hypotenuse is 8√2 times √2, or 16.
POST ALL QUESTIONS IN FULL, INCLUDING ANSWER CHOICES.
in fact, your omission of the answer choices shows that your priorities are all wrong. there are lots of problems on which you can "plug in" - whether plugging in your own numbers or the numbers that actually appear in the answer choices - and use the answer choices to figure out the solution, even if you have no idea how to solve the problem in the "conventional" way.
this is one of those problems, as you'll see below.
if you'd even think of posting a problem without its answer choices, that tells me that you have no idea that such alternate methods exist. if that's the case, then you're needlessly sacrificing a fair # of points from the math section, as such methods are often the easiest way to go.
I know that the ratio is 1:1:sq. root 2, but how do you determine the length of individual sides when given the entire perimeter?
first of all, since this is a multiple-choice question, you don't have to.
you can just plug in an answer choice and work backwards.
since you haven't provided the answer choices, i can only start with the correct answer.
there should be a choice that says 16.
plug this in for the hypotenuse.
in that case, the length of each leg is 16 divided by √2, which is 8√2.
therefore, the perimeter is 2(8√2) + 16, or 16√2 + 16.
this is the correct perimeter, so you're good to go.
done.
(by the way, you should always start with choice (c) when you use this method of substituting the answer choices into the problem. that way, if it's too small, you can eliminate it along with the two smaller choices; if it's too big, you can eliminate it along with the two larger choices.)
--
another thing that can be helpful here is to realize that you can write the ratio not only as 1:1:√2, but also as √2:√2:2,
the perimeter of the former is 2 + 1√2. unfortunately, the given perimeter (16√2) isn't a multiple of this.
the perimeter of the latter is 2 + 2√2. your perimeter is exactly 8 times this value. therefore, your triangle is 8 times the base ratio of √2:√2:2, or 8√2:8√2:16.
--
finally, you can use algebra. the "base" triangle has perimeter 2 + √2, so you can find the desired multiple of this base triangle by dividing your perimeter, 16 + 16√2, by the perimeter of the base triangle, 2 + √2.
that quotient is (16 + 16√2) / (2 + √2) = 8√2. (you have to use the conjugate method to get this.)
therefore, you have 8√2 times the basic 1:1:√2 template, so your hypotenuse is 8√2 times √2, or 16.
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