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# The two sides of a right triangle are 10 square root 2

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Senior RC Moderator
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The two sides of a right triangle are 10 square root 2  [#permalink]

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17 Oct 2018, 10:42
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Difficulty:

45% (medium)

Question Stats:

59% (01:40) correct 41% (02:06) wrong based on 37 sessions

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The two sides of a right triangle are 10$$\sqrt{2}$$ and 5$$\sqrt{2}$$. Which of the following can be the value of the third side?

I. 5$$\sqrt{2}$$
II. II. 5$$\sqrt{6}$$
III. III. 5$$\sqrt{10}$$

A) II only
B) III only
C) I and II only
D) II and III only
E) I, II, and III

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Re: The two sides of a right triangle are 10 square root 2  [#permalink]

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17 Oct 2018, 10:55
Square root of 2 is approx. 1.414

10* 1.414 = 14.14 (1)
5 * 1.414 = 7.07 (2)

The third side is (1) - (2) < x < (1) + (2)

5 sqrt 2 < x < 15 sqrt 2

Sqrt 6 is slightly greater than sqrt 4

Sqrt 10 is slightly greater than sqrt 9

II and III

Answe choice D

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Re: The two sides of a right triangle are 10 square root 2  [#permalink]

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17 Oct 2018, 10:56
Difference of two sides < Third side < Sum of two sides
5√2 < Third side < 15√2 (1)

taking a square of (1)
50 < $$(Third side)^2$$ < 450

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Re: The two sides of a right triangle are 10 square root 2  [#permalink]

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17 Oct 2018, 12:39
1
The two sides of a right triangle are 10$$\sqrt{2}$$ and 5$$\sqrt{2}$$. Which of the following can be the value of the third side?

I. 5$$\sqrt{2}$$
II. 5$$\sqrt{6}$$
III. 5$$\sqrt{10}$$

A) II only
B) III only
C) I and II only
D) II and III only
E) I, II, and III

$$?\,\,\,:\,\,\,3{\rm{rd}}\,\,{\rm{side}}\,\,\,\,\left( {\,{\rm{length}}\,} \right)$$

$$\left. \matrix{ {\left( {10\sqrt 2 } \right)^2} = {\left( {5\sqrt 2 } \right)^2} + {x^2}\,\,\, \Rightarrow \,\,\,\,200 = 50 + {x^2}\,\,\,\,\mathop \Rightarrow \limits^{x\, > \,0} \,\,\,\,x = 5\sqrt 6 \,\,\,\,\, \hfill \cr {y^2} = {\left( {5\sqrt 2 } \right)^2} + {\left( {10\sqrt 2 } \right)^2}\,\,\, \Rightarrow \,\,\,\,{y^2} = 250\,\,\,\,\mathop \Rightarrow \limits^{y\, > \,0} \,\,\,\,y = 5\sqrt {10} \,\, \hfill \cr} \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,?\,\,\,:\,\,\,5\sqrt 6 \,\,{\rm{or}}\,\,5\sqrt {10} \,\,\,\, \Rightarrow \,\,\,\,\left( D \right)$$

Important: in the case of right triangles, the $$a^2=b^2+c^2$$ equality guarantees that the (positive) numbers obtained will be lengths of a viable triangle!
(In short: there is no need to check whether each side is less than the sum of the other two, for instance.)

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: The two sides of a right triangle are 10 square root 2   [#permalink] 17 Oct 2018, 12:39
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