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Geometry - II q3 & q9

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Geometry - II q3 & q9 [#permalink]

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15 Nov 2009, 11:17
Quote:
If points $$A$$ , $$B$$ , and $$C$$ lie on a circle of radius 1, what is the area of triangle $$ABC$$ ?

1. $$AB^2 = BC^2 + AC^2$$
2. $$\angle CAB$$ equals 30 degrees

(C) 2008 GMAT Club - Geometry - II#3

* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
* EACH statement ALONE is sufficient
* Statements (1) and (2) TOGETHER are NOT sufficient
Statements(1) and (2) combined are sufficient. S1 and S2 define the shape of the triangle. Because this triangle can be inscribed in a circle of radius 1, we also know the size of the triangle. There is only one triangle with a given shape and size.

I understand that there's only one possible triangle with that given shape, but how can you know the area?

Also:

Quote:
If the area of a rectangle is 80, what is the angle between the diagonal of the rectangle and its longer side?

1. The perimeter of the rectangle is 84
2. The shorter side of the rectangle is 2

(C) 2008 GMAT Club - Geometry - II#9

* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
* EACH statement ALONE is sufficient
* Statements (1) and (2) TOGETHER are NOT sufficient

Statement (1) by itself is sufficient. We can find the sides of the rectangle (40 and 2).

Statement (2) by itself is sufficient. We can find the sides of the rectangle (40 and 2).

We know the sides but what about the angle between the diagonal and the longer side?
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Re: Geometry - II q3 & q9 [#permalink]

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15 Nov 2009, 13:36
If points $$A$$ , $$B$$ , and $$C$$ lie on a circle of radius 1, what is the area of triangle $$ABC$$ ?

1. $$AB^2 = BC^2 + AC^2$$
2. $$\angle CAB$$ equals 30 degrees

(C) 2008 GMAT Club - Geometry - II#3

* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
* EACH statement ALONE is sufficient
* Statements (1) and (2) TOGETHER are NOT sufficient
Statements(1) and (2) combined are sufficient. S1 and S2 define the shape of the triangle. Because this triangle can be inscribed in a circle of radius 1, we also know the size of the triangle. There is only one triangle with a given shape and size.

I understand that there's only one possible triangle with that given shape, but how can you know the area?

Statement 1 suggests that AB is the diameter of the circle but doesn't specifically tells about the position of C. So area can not be determined. Not Sufficient.

Statement 2 : SUFFIECIENT

Also:

If the area of a rectangle is 80, what is the angle between the diagonal of the rectangle and its longer side?

1. The perimeter of the rectangle is 84
2. The shorter side of the rectangle is 2

(C) 2008 GMAT Club - Geometry - II#9

* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
* EACH statement ALONE is sufficient
* Statements (1) and (2) TOGETHER are NOT sufficient

Statement (1) by itself is sufficient. We can find the sides of the rectangle (40 and 2).

Statement (2) by itself is sufficient. We can find the sides of the rectangle (40 and 2).

We know the sides but what about the angle between the diagonal and the longer side?

Answer D. If you know the sides of a right angle triangle then angle between any of the two sides can be calculated. Since this is a DS question you don't really have to calculate it.

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Re: Geometry - II q3 & q9 [#permalink]

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15 Nov 2009, 14:14
saruba wrote:
Quote:
If points $$A$$ , $$B$$ , and $$C$$ lie on a circle of radius 1, what is the area of triangle $$ABC$$ ?

1. $$AB^2 = BC^2 + AC^2$$
2. $$\angle CAB$$ equals 30 degrees

1 says that the triangle si a right angle triangle.
2 says one of the size of the angle is 30 degree.

1 and 2 say taht the triangle right nagle traingle with angel sizes 30:60:90. Such triangle is possible only with the hypoteneous as diameter. So:

h = d = 2
b (or l) = 1
l (or b) = sqrt3.......Suff

Thats C.
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Re: Geometry - II q3 & q9 [#permalink]

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15 Nov 2009, 14:25
saruba wrote:
If the area of a rectangle is 80, what is the angle between the diagonal of the rectangle and its longer side?

1. The perimeter of the rectangle is 84
2. The shorter side of the rectangle is 2

Statement (1) by itself is sufficient. We can find the sides of the rectangle (40 and 2).

Statement (2) by itself is sufficient. We can find the sides of the rectangle (40 and 2).

We know the sides but what about the angle between the diagonal and the longer side?[/quote]

Given that lb = 80

1: 2(l+b) =84
l+b = 42
80/b + b = 42
l^2 - 42l + 80 = 0
l (l-40) - 40 (l - 40) = 0
(l-40)^2
l = 40
b = 2

With the values of l anf b, we can identify the size of the angle between diagnal and longer side.

How? Need to use triagomatic approach. Our concern is taht if we know the sides of the rectangle, then we can get the size of the desired angle. Suff...

2. If the shorter side of the rectangle is 2, then longer si 40. Suff.

Thats D...
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Re: Geometry - II q3 & q9 [#permalink]

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29 May 2011, 09:18
if $$AB^2 = BC^2 + AC^2$$

is not means that AB is a hypotenuse ?? if yes doesn't it represent the diameter of the circle?
I was thinking that A should be the answer.
can any please correct me .

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Re: Geometry - II q3 & q9 [#permalink]

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29 May 2011, 10:35
pjain01 wrote:
if $$AB^2 = BC^2 + AC^2$$

is not means that AB is a hypotenuse ??
Yes. AB is a hypotenuse.

if yes doesn't it represent the diameter of the circle?
Yes. It does.

I was thinking that A should be the answer.
Many right triangles can be formed with hypotenuse=2. Draw a circle. Considering AB as diameter, and C as any point on the circle, try drawing multiple triangles. Each of these triangles will be right angled triangle with different areas. Thus, this statement is not sufficient to find the area of the triangle.

can any please correct me .

Also discussed here:
area-of-triangle-inside-a-circle-111134.html
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Re: Geometry - II q3 & q9   [#permalink] 29 May 2011, 10:35
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Geometry - II q3 & q9

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