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# The figure above represents a 10-inch piece of string attached to the

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Math Expert
Joined: 02 Sep 2009
Posts: 55264
The figure above represents a 10-inch piece of string attached to the  [#permalink]

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06 Sep 2017, 04:39
00:00

Difficulty:

35% (medium)

Question Stats:

65% (01:21) correct 35% (01:41) wrong based on 29 sessions

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The figure above represents a 10-inch piece of string attached to the ends of a 6-inch rod. If the string is pulled so as to form the sides of a triangle whose base is the rod, then the greatest possible area of such a triangle is

(A) 12
(B) 15
(C) 18
(D) 24
(E) 30

Attachment:

2017-09-06_1535_001.png [ 68.94 KiB | Viewed 826 times ]

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Re: The figure above represents a 10-inch piece of string attached to the  [#permalink]

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06 Sep 2017, 04:57
Length of string =10 inch.

We have to find maximum area of triangle.
max area => max base and max height
Height will be maximum when triangle formed is isoseles

If string is pulled, we will get maximum height when both sides becomes equal. ie Triangle formed = 5,5, 6(base)

Now as triangle formed is isosceles, altitude drawn will be perpendicular bisector of base.
Two triangles formed will be right triangle : with hypotenuse =5 and base =3
So height = root(5^2-3^2) =4

So for triangle base =6, height =4 Area = 1/2 *4* 6 = 12

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Re: The figure above represents a 10-inch piece of string attached to the  [#permalink]

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11 Sep 2017, 10:57
Bunuel wrote:

The figure above represents a 10-inch piece of string attached to the ends of a 6-inch rod. If the string is pulled so as to form the sides of a triangle whose base is the rod, then the greatest possible area of such a triangle is

(A) 12
(B) 15
(C) 18
(D) 24
(E) 30

Attachment:
2017-09-06_1535_001.png

The triangle with the largest possible area is an isosceles triangle with base 6 inches and legs 5 inches each. That is, the 10-inch piece of string is divided into two 5-inch pieces. In this way, the height of the triangle would be the greatest, since the base of the triangle has been fixed to be 6 inches.
We drop a perpendicular to create 2 identical right triangles, each having a base of 3, a hypotenuse of 5, and an unknown height, which we designate as h. Using the Pythagorean theorem, we have:

h^2 + 3^2 = 5^2

h^2 + 9 = 25

h^2 = 16

h = 4

Thus, the largest possible area of the entire triangle is ½ x 6 x 4 = 12 square inches.

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Re: The figure above represents a 10-inch piece of string attached to the   [#permalink] 11 Sep 2017, 10:57
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