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A triangle with side lengths in the ratio 3:4:5 is inscribed in a circ

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A triangle with side lengths in the ratio 3:4:5 is inscribed in a circ  [#permalink]

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14 Mar 2019, 02:49
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55% (hard)

Question Stats:

56% (02:53) correct 44% (02:27) wrong based on 34 sessions

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A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle with radius 3. What is the area of the triangle?

(A) 8.64

(B) 12

(C) $$5\pi$$

(D) 17.28

(E) 18

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A triangle with side lengths in the ratio 3:4:5 is inscribed in a circ  [#permalink]

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Updated on: 15 Mar 2019, 22:51
1
Bunuel wrote:
A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle with radius 3. What is the area of the triangle?

(A) 8.64

(B) 12

(C) $$5\pi$$

(D) 17.28

(E) 18

side ratio 3:4:5
so base = 3 and hypotenuse= diameter = 6
so 5x= 6
x= 6/5
area
0.5 * 3x*4x = 0.5 * 3 * 6/5 * 4* 6/5
IMO A ~ 8. 64

Originally posted by Archit3110 on 14 Mar 2019, 05:16.
Last edited by Archit3110 on 15 Mar 2019, 22:51, edited 1 time in total.
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A triangle with side lengths in the ratio 3:4:5 is inscribed in a circ  [#permalink]

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Updated on: 18 Mar 2019, 05:26
Its a pythogoran triplet so 3*2 6 is the hypotenuse. And the base and height are 18/5 and 24/5. Using the formula we get 8.64. (A)

Posted from my mobile device

Originally posted by Sivaji reddy on 14 Mar 2019, 11:06.
Last edited by Sivaji reddy on 18 Mar 2019, 05:26, edited 1 time in total.
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Re: A triangle with side lengths in the ratio 3:4:5 is inscribed in a circ  [#permalink]

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14 Mar 2019, 11:22
Bunuel wrote:
A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle with radius 3. What is the area of the triangle?

(A) 8.64

(B) 12

(C) $$5\pi$$

(D) 17.28

(E) 18

There are different ways to solve this.

Elimination method (not necessary):
Area circle=9pi=9*22/7
Area half circle=9*11/7=14 1/7.

Eliminate C-E.

Logical approach
Triangle with the largest area, when base is given, is the one with the largest altitude ($$max. altitude=3=r$$).
Area of largest triangle: $$\frac{1}{2}*3*6=9$$. This would be the case if the triangle was a isosceles triangle.
So it has to be smaller than that.

IMO A

Proper method
$$(3x)^2+(4x)^2=(5x)^2$$
Hypotenuse: $$6=5x => x=\frac{6}{5}$$
Area triangle: $$\frac{1}{2}*3x*4x=\frac{1}{2}*3(\frac{6}{5})*4(\frac{6}{5})=\frac{18*12}{25}=\frac{216}{25}=8,...$$

IMO A
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Re: A triangle with side lengths in the ratio 3:4:5 is inscribed in a circ  [#permalink]

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14 Mar 2019, 11:40
1
Something nice to know: whenever you have a right triangle inscribed in a circle, the hypotenuse will match the circle's diameter.

3:4:5 is a right triangle, which means the hypotenuse is equal to the diameter = 2 * radius = 6.

You have, then, the following proportional relationship:

5 is to 4 as 6 is to x; x = 4.8
5 is to 3 as 6 is to y; y = 3.6

Your triangle has, then, the following side lengths: 3.6 - 4.8 - 6.0

The area of a triangle is given by [base x height /2]. Right triangle, so base and height are equal to the two smallest sides:

3.6 * 4.8 / 2 = 8.64

(tip: 3.6 * 4.8 isn't the easiest calculation to make under pressure. 3.6 * 5, however, looks much easier: 18. Divided by 2, you'll have an area that is slightly smaller than 9.

Hope this helps!
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Re: A triangle with side lengths in the ratio 3:4:5 is inscribed in a circ  [#permalink]

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18 Mar 2019, 19:03
Bunuel wrote:
A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle with radius 3. What is the area of the triangle?

(A) 8.64

(B) 12

(C) $$5\pi$$

(D) 17.28

(E) 18

We see that triangle is a right triangle since its side lengths are in the ratio of 3:4:5 or 3x:4x:5x (recall that the 3-4-5 triangle is a right triangle). However, it doesn’t mean that the triangle must be a 3-4-5 triangle or an integer multiple of 3-4-5. It’s possible that sides can be fractional or decimal values, as long as the 3:4:5 ratio is maintained.

Since the triangle is a right triangle and it’s inscribed in a circle, the hypotenuse of the triangle is the diameter of the circle. Therefore, we have:

5x = 2(3)

5x = 6

x = 1.2

Now, the other two sides of the triangle are 3(1.2) = 3.6 and 4(1.2) = 4.8. However, these two sides are the base and height of the triangle, so the area of the triangle is:

(3.6 x 4.8)/2 = 8.64

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Re: A triangle with side lengths in the ratio 3:4:5 is inscribed in a circ   [#permalink] 18 Mar 2019, 19:03
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