Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Is there a rule about area of a triangle in a circle? [#permalink]

Show Tags

19 Apr 2009, 20:27

If a triangle is inscribed in a semi circle, i.e. it is a right angle triangle with diameter as base... is there any rule/formula about finding the area of this triangle?

Also, will all such triangles in a semi circle have the same area? Or the same perimeter?

Re: Is there a rule about area of a triangle in a circle? [#permalink]

Show Tags

19 Apr 2009, 21:53

thinkblue wrote:

If a triangle is inscribed in a semi circle, i.e. it is a right angle triangle with diameter as base... is there any rule/formula about finding the area of this triangle?Sorry, there is no formula such to find the area of the triangle. When we make a triangle in a semicircle.. we are sure about the two things.. 1) the 90degree angle..but the other two angle can be any pair of angle which sum to 90 degree 2) secondly, we are sure about the hypotenuse... But to fix a triangle, we need at least 3 criteria.. So, triangle area cannot be fixed.

Also, will all such triangles in a semi circle have the same area? Or the same perimeter?

No, they dont have the same area or same perimeter if inscribed in a semi-circle.

Re: Is there a rule about area of a triangle in a circle? [#permalink]

Show Tags

21 Apr 2009, 15:16

3

This post received KUDOS

The response that you've already received correctly states that there is no way to know the area of triangle ABD at this point; you'd need some more information. Frankly, that's probably all you wanted!

However, let me add a couple things to this discussion, just in case.

First, a diagram, just to give these things handles:

Thus, line AB is a diameter of the circle with center O, point D is arbitrarily chosen on the circle, and C is straight "up" from the middle (not rigorous, but you know what I mean).

You are correct in saying that ADB must be a right angle, regardless of where D is chosen.

1) There is no lower bound on the area of triangle ADB. If we "push" point D further and further and further to the right, the triangle gets shorter and shorter, see? Thus, the area of triangle ADB can get arbitrarily close to zero.

2) There is, however, an upper bound on the area of triangle ADB! Namely, \(r^2\). When point D coincides with point C, we create a nice 45-45-90 triangle, and visual inspection seems to indicate that it has the maximum area. (the formula is easy to work out: the base of the triangle is the diameter, which is 2r, and the height would then equal r. Thus, \((1/2) * b * h = (1/2) * 2r * r = r^2\)

We can also get that result analytically: Since we're building off of the diameter, the Base of the triangle is fixed. Thus, in order to maximize area, what should you do? Ah, let's maximize the other variable: the Height. And how do you do that? By putting the height right in the middle where it has the most "headroom", so to speak.

Re: Is there a rule about area of a triangle in a circle? [#permalink]

Show Tags

04 May 2009, 12:46

+1 for good explanation.

Liquidhypnotic wrote:

The response that you've already received correctly states that there is no way to know the area of triangle ABD at this point; you'd need some more information. Frankly, that's probably all you wanted!

However, let me add a couple things to this discussion, just in case.

First, a diagram, just to give these things handles:

Thus, line AB is a diameter of the circle with center O, point D is arbitrarily chosen on the circle, and C is straight "up" from the middle (not rigorous, but you know what I mean).

You are correct in saying that ADB must be a right angle, regardless of where D is chosen.

1) There is no lower bound on the area of triangle ADB. If we "push" point D further and further and further to the right, the triangle gets shorter and shorter, see? Thus, the area of triangle ADB can get arbitrarily close to zero.

2) There is, however, an upper bound on the area of triangle ADB! Namely, \(r^2\). When point D coincides with point C, we create a nice 45-45-90 triangle, and visual inspection seems to indicate that it has the maximum area. (the formula is easy to work out: the base of the triangle is the diameter, which is 2r, and the height would then equal r. Thus, \((1/2) * b * h = (1/2) * 2r * r = r^2\)

We can also get that result analytically: Since we're building off of the diameter, the Base of the triangle is fixed. Thus, in order to maximize area, what should you do? Ah, let's maximize the other variable: the Height. And how do you do that? By putting the height right in the middle where it has the most "headroom", so to speak.

Re: Is there a rule about area of a triangle in a circle? [#permalink]

Show Tags

10 May 2009, 12:55

whats the best way to find the diameter of a circle with an inscribed triangle if you are given an arc? so lets say an inscribed triangle with an arc of 24 that is approximately 3/4ths of the circumference. any suggestions?

whats the best way to find the diameter of a circle with an inscribed triangle if you are given an arc? so lets say an inscribed triangle with an arc of 24 that is approximately 3/4ths of the circumference. any suggestions?

Why does the inscribed triangle come into play? I don't see how it adds any helpful information. If there is an Arc and the angle/size is know - it is very easy to find the Diameter.

In your example, the Arc is 3/4th of the circumference, so that means the total circumference is 36.

\(2 * \Pi * R = \Pi * D = 36\), where R is radius and D is diameter

Re: Is there a rule about area of a triangle in a circle? [#permalink]

Show Tags

23 Jul 2013, 07:48

Area: The number of square units it takes to exactly fill the interior of a triangle. Usually called "half of base times height", the area of a triangle is given by the formula below. • A = b*h / 2 Other formula: • A = P*r / 2 • A = abc / 4R Where "b" is the length of the base, "a" and "c" the other sides; "h" is the length of the corresponding altitude; "R" is the Radius of circumscribed circle; "r" is the radius of inscribed circle; "P" is the perimeter

My doubt here is: "P" Is the perimeter of the triangle or the perimeter of the circle?
_________________

Encourage cooperation! If this post was very useful, kudos are welcome "It is our attitude at the beginning of a difficult task which, more than anything else, will affect It's successful outcome" William James

gmatclubot

Re: Is there a rule about area of a triangle in a circle?
[#permalink]
23 Jul 2013, 07:48

Its been long time coming. I have always been passionate about poetry. It’s my way of expressing my feelings and emotions. And i feel a person can convey...

Written by Scottish historian Niall Ferguson , the book is subtitled “A Financial History of the World”. There is also a long documentary of the same name that the...

Post-MBA I became very intrigued by how senior leaders navigated their career progression. It was also at this time that I realized I learned nothing about this during my...