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The shaded region in the figure above represents a frame shaped as a

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Bunuel
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The shaded region in the figure above represents a frame shaped as a [#permalink] New post   13 Sep 2018, 04:50
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The shaded region in the figure above represents a frame shaped as a right triangle with sides 10 inches and 13 inches. The frame encloses a picture, shaped also as a right triangle, which has the same area as the frame itself. If the sum of the 2 legs of the picture is 17 inches, what is the length of the hypotenuse of the triangular picture?


A. \(\sqrt{29}\)

B. \(\sqrt{159}\)

C. 13

D. \(\sqrt{269}\)

E. 17

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The shaded region in the figure above represents a frame shaped as a [#permalink] New post   15 Sep 2018, 02:16
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Frame has the same area as the picture itself means -> Area of ABC – Area of picture = Area of the picture

Area of ABC= 2*area of picture

Area of ABC = (1/2) *10 * 13 = 65

\(65 = 2*1/2(ab)\)

ab=65 -> 130 = 2ab
(a+b) = 17 (GIVEN)

\((a+b)^2 = a^2+b^2+2ab\)
length of the hypotenuse \(c=> c^2=a^2+b^2=289-130=159\)
\(c=\sqrt{159}\) B
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Re: The shaded region in the figure above represents a frame shaped as a [#permalink] New post   13 Sep 2018, 20:57
Hi Bunuel, can you please share the OE or better, your approach? Thanks!
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Re: The shaded region in the figure above represents a frame shaped as a [#permalink] New post   15 Sep 2018, 12:22
Bunuel wrote:

The shaded region in the figure above represents a frame shaped as a right triangle with sides 10 inches and 13 inches. The frame encloses a picture, shaped also as a right triangle, which has the same area as the frame itself. If the sum of the 2 legs of the picture is 17 inches, what is the length of the hypotenuse of the triangular picture?


A. \(\sqrt{29}\)

B. \(\sqrt{159}\)

C. 13

D. \(\sqrt{269}\)

E. 17

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Attachment:
image052.jpg



Given tha the area of the picture is equal to the area of the frame, then

1/2*B*H=(10*13)/2=65.


also area of the inner trianlge(picture)=1/2*b*h=65/2,

this gives b*h=65.


given b+h=17,squaring both sides,


bsquare+hsquare+2bh=289


substituting bh value we get


hypotenus theorm and ans as root 159.


Option B is correct.
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Re: The shaded region in the figure above represents a frame shaped as a [#permalink] New post   28 Oct 2023, 09:41
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Re: The shaded region in the figure above represents a frame shaped as a [#permalink]
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