In the figure, circle O has center O, diameter AB and a radius of 5.

You are not looking at this correctly.

\(1:\sqrt{3}:2\) is only applicable for triangles having angles as \(30^{\circ},60^{\circ}, 90^{\circ}\)

2nd 'special' type of right triangles are the \(45^{\circ},45^{\circ} , 90^{\circ}\) , in which the ratio of the sides becomes \(1:1:\sqrt{2}\)

Right triangles that do not fall under the above 2 cases should not be mixed with the above rules. These triangles have sides that vary with the relative angles inside the triangle. Triangles with sides 3:4:5 or any multiples of this such as 6:8:10 or 9:12:15 etc do not come under the 2 rules mentioned above and thus you will not get the same proportions for the sides.

Hope this clears your doubt

Thanks a lot... That means that 3 - 4 - 5 right triangle is not a 30 - 60 - 90 right triangle?

Yes, exactly. The 3-4-5 triangle or a triangle with any of the other multiples of 3-4-5 is not a 30-60-90 triangle.

FYI, 3-4-5 triangle has angles as \(36.87^{\circ}, 53.13^{\circ},90^{\circ}\) triangle ( For GMAT, you do not have to remember angles that are not 0, 30, 45 60 or 90).

It actually follows from the fact that 30-60-90 triangle will always have sides in the ratio: \(1:\sqrt{3}:2\). The opposite is true as well that if a triangle has sides in the ration \(1:\sqrt{3}:2\), the triangle will then be 30-60-90. The same logic applies to 45-45-90 triangle as well.

Hope this clears your doubt.

Please help me where I am going wrong with this approach.


I got the length of BC =5[square_root]3

Then for calculating the length of the arc I used the angle CBE = 60 with BC and BE.

The length of the arc would be 1/6*2*pi*5[square_root]3

The perimeter = 10[square_root]3 + 1/6*2*pi*5[square_root]3

For calculating the length of the arc, the angle MUST be the center angle (or the angle that the arc makes at the center and NOT at the circumference).

Any angle made by an arc on the circumference = 0.5* Angle made by the SAME arc at the center.

Thus, instead of 60 degrees, you need to calculate the length of arc made by 120 degrees. You will correct answer after that. (1/6) above will become (1/3). Also the entire red equation above is incorrect.

It should be, \(\frac{120}{360} * 2\pi*5 = \frac{10\pi}{3}\)

Thank you for the explanation Engr2012.

But if we consider a circle with the radius of BC and the sector with angle 60, then the arc length could be calculated right?

Look at the attached picture

Yes, if you are given that "x" = 60 (in the attached picture), radius = r , then length of the arc =\(l = r*\theta\). where \(\theta\)= angle in RADIANS and not degrees that is made by the arc at the CENTER. Thus \(\theta\) in this case will be 2*60 = 120 degrees

For degree to radian conversion, use the fact that 180 degrees = \(\pi\) radians.

Thus, for the example you have quoted, \(l = r*[(120/180)*\pi] = r*2*\pi/3\)

Why did we not add length of Diameter AB in the perimeter of the shaded region?

Perimeter is the distance around a figure, or the measurement of the distance around something; the length of the boundary. Since AB is NOT the boundary of the shaded region (it lies within it) we do not add its length when calculating the perimeter.

Hope it's clear.

Due to the fact that in order to calculate perimeter of any area, you only need to consider OUTER bounds of the said area. What is inside, does not matter.

Thus the perimeter for the question asked will be = CB+BE+ minor arc(EAC)

Hope this helps.

Hi, if we were to find the area of the shaded region, what would the solution be.

Thanks

CD and AB are parallel so x = 30 degrees since they are alternate interior angles.

Angle CBE is 2*30 = 60 degrees. This means arc CE subtends a central angle of 120 degrees (Central angle is twice the angle subtended at the circumference).

So arc CE is one third of the circumference of the circle.
Circumference of the circle \(= 2 *\pi * r = 10 * \pi\)

Length of arc CE \(= (10/3) * \pi\)

Since angle CBE is 60 degrees, triangle CBE is equilateral.

The ratio of side of the equilateral triangle inscribed in a circle and the radius of the circle \(= a : r = \sqrt{3}:1\)
So \(BC = BE = 5*\sqrt{3}\)

\(Perimeter = (10/3) * \pi + 10*\sqrt{3}\)

Answer (D)

IMPORTANT: unless stated otherwise, the diagrams in Problem Solving geometry questions are DRAWN TO SCALE.

I'll solve this question using estimation.

Since the diameter AB = 10, we can ESTIMATE the length of CB.
It looks like CB is just a little bit shorter than AB.
So, I'll say that the length of side CB is approximately 9.
This means the length of side EB is approximately 9 as well.
Finally, arc EC looks a little bit shorter than sides CB and EB, so I'll estimate it to be length 8

So, the TOTAL perimeter = 9 + 9 + 8 = 26

Now check the answer choices:

ASIDE: On test day, everyone should know the following apprximations:
√2 ≈ 1.4
√3 ≈ 1.7
√5 ≈ 2.2
Also, we'll say that pi ≈ 3

A. (5/3)pi + 5√3 ≈ 5 + 8.5 ≈ 13.5
B. (5/3)pi + 10√3 ≈ 5 + 17 ≈ 22
C. (10/3)pi + 5√3 ≈ 10 + 8.5 ≈ 18.5
D. (10/3)pi + 10√3 ≈ 10 + 17 ≈ 27
E. (10/3)pi + 20√3 ≈ 10 + 34 ≈ 44

Of these, it appears that D is the closest.

Aside: We can see that answer choice B is pretty close too. At this point, you have a time-management decision. You can either stick with D, and use your extra time elsewhere, or your can spend time trying to be more certain of the answer. Your choice.

That said, D is the correct answer.

Cheers,
Brent

<x = alt. <BCD = 30 degrees.
<CBE = 60 degree

O be the center and the angle subtended at the center = <COE = 120 degree

Length of the arc(CAE) = (120/360)*pi*5^2

Now considering triangle OCB, OB = OC = 5 units.
<COB = 120 degree
5/sin30 = BC/sin120
BC = 5sqrt3

Total perimeter = (120/360)*pi*5^2 + 10sqrt3 (D)

Could someone provide insight about what this would look like drawn out and labeled? I am trying to grasp it. Thank you so much!

Hello Bunuel,

I imagined that point B is the centre of another circle say P.
now we know the radius of the circle as BA; now BC = BE = 10
From the figure, we know that x = 30

we have a radius and an angle, we can find arc length.

the answer should be (10pi/3) + 20.

please clarify where am I missing.

Thank you.

I think the figure below might help:

So, if B is the center and BA is the radius, then you'd get above circle. For that you can see why your further logic is not correct.