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QOTD # 9 The length of a leg of an isosceles right triangle

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QOTD # 9 The length of a leg of an isosceles right triangle [#permalink] New post 13 Sep 2016, 05:52
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Question Stats:

61% (01:35) correct 38% (00:46) wrong based on 52 sessions
Quantity A
Quantity B
The length of a leg of an isosceles right triangle with area R
The length of a side of a square with area R



A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.

Practice Questions
Question: 9
Page: 204
[Reveal] Spoiler: OA

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Re: QOTD # 9 The length of a leg of an isosceles right triangle [#permalink] New post 13 Sep 2016, 05:55
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Explanation


In an isosceles right triangle, both legs have the same length. The area of an isosceles right triangle with legs of length x is equal to \(\frac{x^2}{2}\) The area of a square is \(s^2\) . Since you are given that an isosceles right triangle and a square have the same area R, it follows that \(\frac{x^2}{2}=s^2\) and so \(x=\sqrt{2}s\)

Since \(\sqrt{2}\) is greater than 1, the length of a leg of the triangle is greater than the length of a side of the square.

The correct answer is Choice A
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Re: QOTD # 9 The length of a leg of an isosceles right triangle [#permalink] New post 12 Apr 2018, 08:58
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Carcass wrote:
Quantity A
Quantity B
The length of a leg of an isosceles right triangle with area R
The length of a side of a square with area R



A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.


The length of a leg of an isosceles right triangle with area R
Let x = the length of one leg
Since this is an isosceles right triangle, the length of the other leg must also be x
Also, since this is a right triangle, then the base has length x AND the height is x

Area of triangle = (base)(height)/2
So, we can write: R = (x)(x)/2
Simplify: R = x²/2
Multiply both sides by 2 to get: 2R = x²
Take square root of both sides to get: √(2R) = x

Great, we have now written Quantity A in terms of R

The length of a side of a square with area R
Let y = the length of one side of square
So, we can write: R = (y)(y)
Simplify: R = y²
Take square root of both sides to get: √R = y

We have now written Quantity B in terms of R

So, we now have the following:
Quantity A: √(2R)
Quantity B: √R

At this point, we may recognize that Quantity A is greater.
However, if you need more convincing, we can take √(2R) and REWRITE it as (√2)(√R) to get:
Quantity A: (√2)(√R)
Quantity B: √R

Now divide both quantities by √R to get:
Quantity A: √2
Quantity B: 1

Since √2 ≈ 1.4, we can see that Quantity A is greater

Answer: A

Cheers,
Brent
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Re: QOTD # 9 The length of a leg of an isosceles right triangle [#permalink] New post 12 Apr 2018, 09:49
When the question says one of the legs of the triangle, does that mean it is one of the 2 equal sides of the isosceles triangle? Can it not be the 3rd side?
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Re: QOTD # 9 The length of a leg of an isosceles right triangle [#permalink] New post 12 Apr 2018, 09:54
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mohan514 wrote:
When the question says one of the legs of the triangle, does that mean it is one of the 2 equal sides of the isosceles triangle? Can it not be the 3rd side?


Good question.

In a right triangle, the two sides that form the 90-degree angle are called legs.
The remaining (and longest) side is called the hypotenuse.

Cheers,
Brent
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Re: QOTD # 9 The length of a leg of an isosceles right triangle [#permalink] New post 15 Feb 2019, 06:33
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Carcass wrote:
Quantity A
Quantity B
The length of a leg of an isosceles right triangle with area R
The length of a side of a square with area R



A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.

Practice Questions
Question: 9
Page: 204


We can also think about this question visually with less calculation.

If an isosceles right triangle have equal area with a square, can the square's side be equal to the triangle's leg?

If the leg of an isosceles right triangle is equal to a square's side, then the square must have twice the area of the triangle, since a diagonal can divide the square into 2 equal isosceles right triangle with legs equal to the square's side.

So in order for the triangle to have same area as the square it must have a larger leg length than the side of the square.

Hope I was clear. :)
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Re: QOTD # 9 The length of a leg of an isosceles right triangle [#permalink] New post 21 Feb 2019, 22:47
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no need to do too much triangle identity work here.
area of triangle =1/2*b*h since it's isoceles, b=h, then...
1/2*b^2=R
b^2=2R

area of the square
each leg is a base
b^2=R

we see that the b^2 of the triangle is 2R
while the b^2 of the square is just R

Thus we choose A
Re: QOTD # 9 The length of a leg of an isosceles right triangle   [#permalink] 21 Feb 2019, 22:47
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QOTD # 9 The length of a leg of an isosceles right triangle

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