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Math Expert V
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If a triangle inscribed in a semicircle has area 40, what is the area  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 27% (02:22) correct 73% (01:25) wrong based on 44 sessions

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If a triangle inscribed in a semicircle has area 40, what is the area of the circle?

(1) One side of the triangle is equal to the diameter of the circle.
(2) The measure of one of the angles in the triangle is 30.

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If a triangle inscribed in a semicircle has area 40, what is the area  [#permalink]

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Bunuel wrote:
If a triangle inscribed in a semicircle has area 40, what is the area of the circle?

(1) One side of the triangle is equal to the diameter of the circle.
(2) The measure of one of the angles in the triangle is 30.

Question stem:- we need the radius of circle in order to determine area of circle.
Given , a triangle is inscribed in a semicircle . By virtue of its property, 2 of the 3 vertices of the traingle lie on the extreme points of the diameter of the semicircle. Therefore , st1 is a repeat of the inherent property of *a triangle inscribed in a semicircle*
Hence st1 insufficient.

St2:- The triangle is a special right angled triangle of the form 30-60-90.

Area of the triangle is given. Hence radius of the semicircle can be calculated.
Sufficient.
(Sides of the triangle are in the ratio r:√3r:2r. Hence area=1/2*r*√3r=40. Value of r can be determined,actual computation is a waste of precious time)

Ans. (B)
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Re: If a triangle inscribed in a semicircle has area 40, what is the area  [#permalink]

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PKN wrote:
Bunuel wrote:
If a triangle inscribed in a semicircle has area 40, what is the area of the circle?

(1) One side of the triangle is equal to the diameter of the circle.
(2) The measure of one of the angles in the triangle is 30.

Question stem:- we need the radius of circle in order to determine area of circle.
Given , a triangle is inscribed in a semicircle . By virtue of its property, 2 of the 3 vertices of the traingle lie on the extreme points of the diameter of the semicircle. Therefore , st1 is a repeat of the inherent property of *a triangle inscribed in a semicircle*
Hence st1 insufficient.

St2:- The triangle is a special right angled triangle of the form 30-60-90.

Area of the triangle is given. Hence radius of the semicircle can be calculated.
Sufficient.
(Sides of the triangle are in the ratio r:√3r:2r. Hence area=1/2*r*√3r=40. Value of r can be determined,actual computation is a waste of precious time)

Ans. (B)

Hi,

From statement 2, aren't you assuming that the 3 angle measures are 30, 60 and 90? The statement mentions 1 angle is 30, the others could be of any combination summing up to 150.
I think only when we combine stmt 1 and 2 we can derive that its a right triangle and thus the angles would be 30/60/90 and eventually calculate the diameter and area of the circle.

Should be C in my view.
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If a triangle inscribed in a semicircle has area 40, what is the area  [#permalink]

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Shruti0805 wrote:
PKN wrote:
Bunuel wrote:
If a triangle inscribed in a semicircle has area 40, what is the area of the circle?

(1) One side of the triangle is equal to the diameter of the circle.
(2) The measure of one of the angles in the triangle is 30.

Question stem:- we need the radius of circle in order to determine area of circle.
Given , a triangle is inscribed in a semicircle . By virtue of its property, 2 of the 3 vertices of the traingle lie on the extreme points of the diameter of the semicircle. Therefore , st1 is a repeat of the inherent property of *a triangle inscribed in a semicircle*
Hence st1 insufficient.

St2:- The triangle is a special right angled triangle of the form 30-60-90.

Area of the triangle is given. Hence radius of the semicircle can be calculated.
Sufficient.
(Sides of the triangle are in the ratio r:√3r:2r. Hence area=1/2*r*√3r=40. Value of r can be determined,actual computation is a waste of precious time)

Ans. (B)

Hi,

From statement 2, aren't you assuming that the 3 angle measures are 30, 60 and 90? The statement mentions 1 angle is 30, the others could be of any combination summing up to 150.
I think only when we combine stmt 1 and 2 we can derive that its a right triangle and thus the angles would be 30/60/90 and eventually calculate the diameter and area of the circle.

Should be C in my view.

HiShruti0805 ,

Why st1 is insufficient is already explained.

Refer red-highlightes line in your post. It's incorrect, rather st2 says one of the angle of the triabgle is 30 degree.

You know, any diameter of a circle subtends a right angle to any point on the circle.
Now we know two angles of the triangle( 90 and 30 degree) , so th emeasure of the third side is 180-90-30=60 degree
Hence the triangle is in the form 30-60-90.

Hope it's clear.
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Re: If a triangle inscribed in a semicircle has area 40, what is the area  [#permalink]

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PKN wrote:
Shruti0805 wrote:
PKN wrote:
[quote="Bunuel"]If a triangle inscribed in a semicircle has area 40, what is the area of the circle?

(1) One side of the triangle is equal to the diameter of the circle.
(2) The measure of one of the angles in the triangle is 30.

Question stem:- we need the radius of circle in order to determine area of circle.
Given , a triangle is inscribed in a semicircle . By virtue of its property, 2 of the 3 vertices of the traingle lie on the extreme points of the diameter of the semicircle. Therefore , st1 is a repeat of the inherent property of *a triangle inscribed in a semicircle*
Hence st1 insufficient.

St2:- The triangle is a special right angled triangle of the form 30-60-90.

Area of the triangle is given. Hence radius of the semicircle can be calculated.
Sufficient.
(Sides of the triangle are in the ratio r:√3r:2r. Hence area=1/2*r*√3r=40. Value of r can be determined,actual computation is a waste of precious time)

Ans. (B)

Hi,

From statement 2, aren't you assuming that the 3 angle measures are 30, 60 and 90? The statement mentions 1 angle is 30, the others could be of any combination summing up to 150.
I think only when we combine stmt 1 and 2 we can derive that its a right triangle and thus the angles would be 30/60/90 and eventually calculate the diameter and area of the circle.

Should be C in my view.

HiShruti0805 ,

Why st1 is insufficient is already explained.

Refer red-highlightes line in your post. It's incorrect, rather st2 says one of the angle of the triabgle is 30 degree.

You know, any diameter of a circle subtends a right angle to any point on the circle.
Now we know two angles of the triangle( 90 and 30 degree) , so th emeasure of the third side is 180-90-30=60 degree
Hence the triangle is in the form 30-60-90.

Hope it's clear.[/quote]Hi,

Statement 2 doesn't really mention that the entire diameter is being used as a side. We could use just a smaller part of the diameter as a side and form a triangle that doesn't have its 3rd vertex on the edge of the circle, right? That creates a triangle which isn't a right triangle and makes this statement insufficient.

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If a triangle inscribed in a semicircle has area 40, what is the area  [#permalink]

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Quote:

Statement 2 doesn't really mention that the entire diameter is being used as a side. We could use just a smaller part of the diameter as a side and form a triangle that doesn't have its 3rd vertex on the edge of the circle, right? That creates a triangle which isn't a right triangle and makes this statement insufficient.

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Hi,

What do we mean by the term INSCRIBED?
it usually means drawing one shape inside another so that it just touches.

HERE THE TRIANGLE IS INSCRIBED IN THE SEMI-CIRCLE, which implies that the vertices of the triangle touch the circumference of the semi-circle and two extreme points of the diameter. If you consider smaller portion of the diameter as the sides of the triangle then we can't call the resultant figure as an inscribed triangle in a semi-circle.

A sample figure is enclosed.
Attachments semi-circle.PNG [ 28.11 KiB | Viewed 428 times ]

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Re: If a triangle inscribed in a semicircle has area 40, what is the area  [#permalink]

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1
1

Solution

Given:
• A triangle, which is inscribed in a semicircle, has area of 40 sq. units

To find:
• Area of the circle

Analysing Statement 1
“One side of the triangle is equal to the diameter of the circle”

• Implies, it is a right-angled triangle
• And, the hypotenuse of the triangle is the diameter of the circle
• However, we cannot find the diameter, as we do not have any information about the other sides of the triangle.

Therefore, Statement 1 is NOT sufficient to arrive at a unique answer

Analysing Statement 2
“The measure of one of the angles is 30”

• The other angles of the triangle can be anything, such that their sum = 180 – 30 = 150
• However, we cannot infer anything else from this information

Therefore, Statement 2 is NOT sufficient to arrive at a unique answer

Combining Both Statements
• From Statement 1, we know that, one of the angles of triangle = 90, and the diameter of the circle is the hypotenuse of the triangle
• From Statement 2, we know that one of the angles of triangle = 30
• Combining both, we get, the angles of the triangle as 30. 60 and 90
• Now, let’s assume the diameter as D
o The other two sides of the triangle will be, DSin30 and DCos30 = D/2 and √3D/2
o Thus, area of the triangle = 1/2 * D/2 * √3D/2 = 40
• From this we get a unique value of D, so, we get a unique value for the area of the circle.

Therefore, both statements TOGETHER are sufficient to arrive at a unique answer.

Hence, the correct answer is option C.

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Re: If a triangle inscribed in a semicircle has area 40, what is the area  [#permalink]

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Bunuel & EgmatQuantExpert:

OA provided by Bunuel is different from the Answer derived by EgmatQuantExpert.

Can we have a consensus on this, plz?
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Re: If a triangle inscribed in a semicircle has area 40, what is the area  [#permalink]

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RLokesh wrote:
Bunuel & EgmatQuantExpert:

OA provided by Bunuel is different from the Answer derived by EgmatQuantExpert.

Can we have a consensus on this, plz?

good catch I am also looking for clarification Re: If a triangle inscribed in a semicircle has area 40, what is the area   [#permalink] 22 Sep 2018, 13:51
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