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

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Math Expert
Joined: 02 Sep 2009
Posts: 55648
The shaded region in the figure above represents a frame shaped as a  [#permalink]

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13 Sep 2018, 04:50
00:00

Difficulty:

65% (hard)

Question Stats:

50% (03:08) correct 50% (03:31) wrong based on 36 sessions

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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

Attachment:

image052.jpg [ 4.88 KiB | Viewed 733 times ]

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Re: The shaded region in the figure above represents a frame shaped as a  [#permalink]

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13 Sep 2018, 20:57
Hi Bunuel, can you please share the OE or better, your approach? Thanks!
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The shaded region in the figure above represents a frame shaped as a  [#permalink]

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15 Sep 2018, 02:16

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]

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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

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.
Re: The shaded region in the figure above represents a frame shaped as a   [#permalink] 15 Sep 2018, 12:22
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