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Inscribed Triangles.gif [ 2.27 KiB | Viewed 10010 times ]

In the figure, ABC is an equilateral triangle, and DAB is a right triangle. What is the area of the circumscribed circle?

You should know the following properties to solve this question: • All inscribed angles that subtend the same arc are equal. The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Hence, all inscribed angles that subtend the same arc are equal. • A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle. • In a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio \(1 : \sqrt{3}: 2\). Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°). For more check Circles Triangles and chapters of Math Book: math-circles-87957.html and math-triangles-87197.html

So, from above we'll have that as DAB=90 degrees then DB must be a diameter of the circle. Next, as angles ACB and ADB subtend the same arc AB then they must be equal and since ACB=60 (remeber ACB is an equilateral triangle) then ADB=60 too. Thus DAB is 30-60-90 triangle and its sides are in ratio \(1 : \sqrt{3}: 2\).

(1) DA = 4 --> the side opposite 30 degrees is 4, then hypotenuse DB=diameter=4*2=8 --> radius=4 --> \(area=\pi{r^2}=16\pi\). Sufficient.

(2) Angle ABD = 30 degrees --> we knew this from the stem, so nothing new. Not sufficient.

Merging similar topics. Please ask if anything remains unclear.

Thanks for merging! I don't get why the Central Angle Theorem applies. In the figure, there is no central angle for that to occur with. So why would ADB be the same as ACB?

Merging similar topics. Please ask if anything remains unclear.

Thanks for merging! I don't get why the Central Angle Theorem applies. In the figure, there is no central angle for that to occur with. So why would ADB be the same as ACB?

All inscribed angles that subtend the same arc are equal. Since angles ACB and ADB subtend the same arc AB then they must be equal. Next, there is no central angle shown on the diagram, I just mentioned Central Angle Theorem to explain why is above property true: angles ACB and ADB have the same central angle AOB (O is the center of the circle) and since they both equal to half of it then they must be equal.

So I chose both statements together are sufficient but here is the explanation provided: Since both angles ACB and ADB are opposite side AB, and angle ACB is part of an equilateral triangle, we can say both ACB and ADB are 60 degrees.

Why does that hold true? I get that : 1. DAB is 90 2. CAB = ACB = BAC = 60

and based on that all i can say is ABD < 60 so if we arent told ABD's value, cant ABD = 70 and ADB = 20?

In any case, if we can deduce ABD from just statement 2, then the rest is easy. if ABD = 30 degrees (as told by statement 1), then DB = diameter. we can figure out the length of DB using the 30-60-90 relationship and solve for area.

So I chose both statements together are sufficient but here is the explanation provided: Since both angles ACB and ADB are opposite side AB, and angle ACB is part of an equilateral triangle, we can say both ACB and ADB are 60 degrees.

Why does that hold true? I get that : 1. DAB is 90 2. CAB = ACB = BAC = 60

and based on that all i can say is ABD < 60 so if we arent told ABD's value, cant ABD = 70 and ADB = 20?

In any case, if we can deduce ABD from just statement 2, then the rest is easy. if ABD = 30 degrees (as told by statement 1), then DB = diameter. we can figure out the length of DB using the 30-60-90 relationship and solve for area.

thanks.

There is a thing you are not considering: in any right triangle inscribed in a circle, the hypotenuse coincides with a diameter of the circle. So we know that DB is the diameter, that the bisectors of the ACB triangle intersect at the center (ancient math theory if I remember...). ABC=60° is divided in two 30°angles, DAB=90° and so ADB=60° And now as you said "we can figure out the length of DB using the 30-60-90 relationship and solve for area."
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This is very good trap C DS questions and if running out of time on exam you want to guess anything but C.

That said, the problem is using the relationship between an inscribed angle and central angle. central angle is the angle formed by an arc at the center and inscribed angle is the angle formed by the same arc at the circle.

central angle = 2 * inscribed angle - Here <ACB and <ADB are inscribed angles formed by arc AB.

From the stem, it is given that <ACB = 60 so the central angle for arch AB is 120. Make O as the center of the circle, so <AOB = 120

Now <ADB is the inscribed angle for <AOB so it will be 60 degrees. Triangle ADB is a 30-60-90 type right angle triangle and stmt 2 gives DA = 8 using which you can calculate the diameter of the circle as 16 and circumference as 16pi. Hence stmt B is sufficient.

Thanks for sharing this problem...

//Kudos please, if the above explanation is good.
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This is very good trap C DS questions and if running out of time on exam you want to guess anything but C.

That said, the problem is using the relationship between an inscribed angle and central angle. central angle is the angle formed by an arc at the center and inscribed angle is the angle formed by the same arc at the circle.

central angle = 2 * inscribed angle - Here <ACB and <ADB are inscribed angles formed by arc AB.

From the stem, it is given that <ACB = 60 so the central angle for arch AB is 120. Make O as the center of the circle, so <AOB = 120

Now <ADB is the inscribed angle for <AOB so it will be 60 degrees. Triangle ADB is a 30-60-90 type right angle triangle and stmt 2 gives DA = 8 using which you can calculate the diameter of the circle as 16 and circumference as 16pi. Hence stmt B is sufficient.

Thanks for sharing this problem...

//Kudos please, if the above explanation is good.

Also did not consider this central angle-inscribed angle relationship... very interesting

I came across a DS question whose explanation provides reasoning I do not follow. What am i missing?

We know that angle subtended by the same arc, on the circumference are always equal.

From F.S 1, the data given is of no value. It is so because Angle ACB = Angle ADB = 60 degrees. Thus, angle ABD has to be (90-60) = 30 degrees.Insufficient.

From F.S 2, we know the length of AD. Thus, in the case of a 60-30-90 triangle, we can easily calculate the hypotenuse = the diameter =BD and hence the area.Sufficient.

Re: In the figure, ABC is an equilateral triangle, and DAB is [#permalink]

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11 Oct 2015, 09:22

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