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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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New post 14 Mar 2019, 02:49
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Question Stats:

52% (03:11) correct 48% (02:21) wrong based on 25 sessions

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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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New post Updated on: 15 Mar 2019, 22:51
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
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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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New post 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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New post 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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New post 14 Mar 2019, 11:40
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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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New post 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

Answer: A
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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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