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Inscribed Triangle

by yserious » Thu Dec 24, 2009 5:56 am
For the triangle shown above, where A, B and C are all points on a circle, and line segment AB has length 18, what is the area of triangle ABC?

(1) Angle ABC measures 30°.

(2) The circumference of the circle is 18*PI.

In order to find the area of the triangle, we need to find the lengths of a base and its associated height. Our strategy will be to prove that ABC is a right triangle, so that CB will be the base and AC will be its associated height.
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OA is C..Not sure why its not B
Thought that there is a property " Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. "
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by linkinpark » Thu Dec 24, 2009 6:14 am
yserious wrote:
OA is C..Not sure why its not B
Thought that there is a property " Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. "
from stmt2, we can assure AB is diameter of circle but what about other sides?not sufficient
the property you've mentioned is true but without knowing stmt 1 how will you find the other two angles of triangle, from stmt1 and 2 together we come to know its 30,60,90 trianlge i.e. we could use sine and cosine to find the other side and hence area.
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by yserious » Thu Dec 24, 2009 6:24 am
linkinpark wrote:
yserious wrote:
OA is C..Not sure why its not B
Thought that there is a property " Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. "
from stmt2, we can assure AB is diameter of circle but what about other sides?not sufficient
the property you've mentioned is true but without knowing stmt 1 how will you find the other two angles of triangle, from stmt1 and 2 together we come to know its 30,60,90 trianlge i.e. we could use sine and cosine to find the other side and hence area.
oops..how could i forget that...thanks linkinpark:)your right
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by tata » Thu Dec 24, 2009 11:14 am
yserious wrote:For the triangle shown above, where A, B and C are all points on a circle, and line segment AB has length 18, what is the area of triangle ABC?

(1) Angle ABC measures 30°.

(2) The circumference of the circle is 18*PI.

In order to find the area of the triangle, we need to find the lengths of a base and its associated height. Our strategy will be to prove that ABC is a right triangle, so that CB will be the base and AC will be its associated height.
Image


OA is C..Not sure why its not B
Thought that there is a property " Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. "

I think the answer is A

Statement 1. Angle ABC is 30 degrees, so this is a 30-60-90 triangle making sides as 1:SQRT3:2, we know AB = 18, we can find out AC and AB(no need to calculate) and hence we can calculate the area of the triangle. - SUFFICIENT

Statement2. Circumference is 18*PI, which gives us radius as 9, that is already mentioned in the stem. NOT SUFFICIENT

IMO A is the OA
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by linkinpark » Thu Dec 24, 2009 11:26 am
tata wrote:
yserious wrote:For the triangle shown above, where A, B and C are all points on a circle, and line segment AB has length 18, what is the area of triangle ABC?

(1) Angle ABC measures 30°.

(2) The circumference of the circle is 18*PI.

In order to find the area of the triangle, we need to find the lengths of a base and its associated height. Our strategy will be to prove that ABC is a right triangle, so that CB will be the base and AC will be its associated height.
Image


OA is C..Not sure why its not B
Thought that there is a property " Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. "

I think the answer is A

Statement 1. Angle ABC is 30 degrees, so this is a 30-60-90 triangle making sides as 1:SQRT3:2, we know AB = 18, we can find out AC and AB(no need to calculate) and hence we can calculate the area of the triangle. - SUFFICIENT

Statement2. Circumference is 18*PI, which gives us radius as 9, that is already mentioned in the stem. NOT SUFFICIENT

IMO A is the OA
I think you're guessing based on the given drawing, but don't assume anything unless it's mentioned implicitly or it can be drawn out from given information.
in A it says angle is 30 but we don't know yet if the triangle is right angled so its insufficient.
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by yserious » Thu Dec 24, 2009 12:42 pm
OA is C
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by Stuart@KaplanGMAT » Fri Dec 25, 2009 2:32 am
linkinpark wrote:
I think you're guessing based on the given drawing, but don't assume anything unless it's mentioned implicitly or it can be drawn out from given information.
in A it says angle is 30 but we don't know yet if the triangle is right angled so its insufficient.
This is an incredibly important principle for data sufficiency: don't assume anything (well, almost nothing).

One key difference between PS and DS is geometry diagrams.

In PS, diagrams are, by default, drawn to scale; unless they specifically state otherwise, you can rely on how diagrams look to help you answer questions.

In DS, on the other hand, diagrams are NOT drawn to scale; we can only rely on information that we're specifically given.

So, in this particular question, it looks like AB is a diameter of the circle, but until we're explicitly (or implicitly) told that AB is the diameter, it's just a random chord of the circle.

Once we know it's the diameter from (2), we know that ABC is a right triangle (any triangle formed inside the circle by the diameter and a point on the circumference will be right). However, by itself that's not enough info, since there are an infinite number of different right triangles we can form with the diameter as the hypotenuse.

Adding (1) into the equation, we know we have a 30/60/90 triangle and we know one of the sides, so we can figure out whatever we want about the triangle.
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by mahendra700 » Fri Dec 25, 2009 3:31 am
For this question just remember the principle: If we know length of one side of the triangle & one angle of three angles then we can find area of a triangle(of course it should be right angle triangle)....
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by Stuart@KaplanGMAT » Fri Dec 25, 2009 3:43 am
mahendra700 wrote:For this question just remember the principle: If we know length of one side of the triangle & one angle of three angles then we can find area of a triangle(of course it should be right angle triangle)....
That principle is NOT correct. Just knowing 1 side and 1 angle is not sufficient to describe a specific triangle.

If you know 1 side and 2 angles (which of course also tells us the third angle), and know where the side is relative to the 2 angles, then you know everything about the triangle.
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by Khaledalyawad » Sat Aug 06, 2011 12:40 am
Answer should be A not C as we have the angle abc known (30) and the hypotenuse is (18) using the cos(angle) = adjacent / hypotenuse we will be able to get the adjacent then using the sin function we will be able to get the other angles and and so on till we calculate the area
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by shivanigs » Wed Oct 31, 2012 2:40 am
Hi,

This is an old post,but Stuart I would be most grateful if you could please help me understand the following:

Statement 1 tells us that angle abc is 30 degrees,so why can we not drop a perpendicular bisector down from c to side ab and convert it into a 30 - 60 -60 triangle and thereby find the area.

Awaiting your response.

Regards..
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by Anaira Mitch » Fri Dec 30, 2016 2:36 am
In order to find the area of the triangle, we need to find the lengths of a base and its associated height. Our strategy will be to prove that ABC is a right triangle, so that CB will be the base and AC will be its associated height.

(1) INSUFFICIENT: We now know one of the angles of triangle ABC, but this does not provide sufficient information to solve for the missing side lengths.

(2) INSUFFICIENT: Statement (2) says that the circumference of the circle is 18 . Since the circumference of a circle equals times the diameter, the diameter of the circle is 18. Therefore AB is a diameter. However, point C is still free to "slide" around the circumference of the circle giving different areas for the triangle, so this is still insufficient to solve for the area of the triangle.

(1) AND (2) SUFFICIENT: Note that inscribed triangles with one side on the diameter of the circle must be right triangles. Because the length of the diameter indicated by Statement (2) indicates that segment AB equals the diameter, triangle ABC must be a right triangle. Now, given Statement (1) we recognize that this is a 30-60-90 degree triangle. Such triangles always have side length ratios of 1: sqrt3 :2

Given a hypotenuse of 18, the other two segments AC and CB must equal 9 and 9sqrt3 respectively. This gives us the base and height lengths needed to calculate the area of the triangle, so this is sufficient to solve the problem.

The correct answer is C.
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by Jay@ManhattanReview » Thu Jan 19, 2017 10:55 pm
shivanigs wrote:Hi,

This is an old post,but Stuart I would be most grateful if you could please help me understand the following:

Statement 1 tells us that angle abc is 30 degrees,so why can we not drop a perpendicular bisector down from c to side ab and convert it into a 30 - 60 -60 triangle and thereby find the area.

Awaiting your response.

Regards..
Your thinking is correct. Drop a perpendicular from point C on chord AB, making a 30 - 60 -60 triangle. Once we get the value of the base, we get the value of height (perpendicular). Say the perpendicular from point C meets AB at point D. Thus, CD = height. But the fact is: we do not know the length of BD (the base of triangle BCD). Thus, statement 1 itself is not sufficient.

Hope this helps!

-Jay
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