GMAT Math : Calculating the angle of a sector

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Example Questions

Example Question #1 : Calculating The Angle Of A Sector

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Note: Figure NOT drawn to scale.

\(\displaystyle m \angle A = 65 ^{\circ }; m \angle B = 55 ^{\circ }\).

Order the degree measures of the arcs \(\displaystyle \widehat{AB}, \widehat{BC}, \widehat{CA}\) from least to greatest.

Possible Answers:

\(\displaystyle \widehat{ BC}, \widehat{ AB} , \widehat{ AC }\)

\(\displaystyle \widehat{ AB} , \widehat{ AC } , \widehat{ BC}\)

\(\displaystyle \widehat{ AB} , \widehat{ BC}, \widehat{ AC }\)

\(\displaystyle \widehat{ AC} , \widehat{ BC } , \widehat{ AB}\)

\(\displaystyle \widehat{ AC} , \widehat{ AB } , \widehat{ BC}\)

Correct answer:

\(\displaystyle \widehat{ AC} , \widehat{ AB } , \widehat{ BC}\)

Explanation:

\(\displaystyle m \angle C = 180 ^{\circ } - m \angle A - m \angle B = 180 ^{\circ } - 65^{\circ } - 55^{\circ } = 60^{\circ }\)

\(\displaystyle m \angle B < m \angle C < m \angle A\), so, by the Multiplication Property of Inequality,

\(\displaystyle 2 \cdot m \angle B < 2 \cdot m \angle C < 2 \cdot m \angle A\).

The degree measure of an arc is twice that of the inscribed angle that intercepts it, so the above can be rewritten as 

\(\displaystyle m \widehat{ AC} < m\widehat{ AB } < m\widehat{ BC}\).

Example Question #21 : Geometry

In the figure shown below, line segment \(\displaystyle AB\) passes through the center of the circle and has a length of \(\displaystyle 8\:cm\). Points \(\displaystyle A\)\(\displaystyle B\), and \(\displaystyle C\) are on the circle. Sector \(\displaystyle COB\) covers \(\displaystyle \frac{1}{6}th\) of the total area of the circle. Answer the following questions regarding this shape.

Circle1

Find the value of central angle \(\displaystyle AOC\).

Possible Answers:

\(\displaystyle 60^{\circ}\)

\(\displaystyle 150^{\circ}\)

\(\displaystyle 240^{\circ}\)

\(\displaystyle 180^{\circ}\)

\(\displaystyle 120^{\circ}\)

Correct answer:

\(\displaystyle 120^{\circ}\)

Explanation:

Here we need to recall the total degree measure of a circle. A circle always has exactly \(\displaystyle 360\) degrees. 

Knowing this, we need to utilize two other clues to find the degree measure of \(\displaystyle AOC\).

1) Angle \(\displaystyle AOB\) measures \(\displaystyle 180\) degrees, because it is made up of line segment \(\displaystyle AB\), which is a straight line.

2) Angle \(\displaystyle BOC\) can be found by using the following equation. Because we are given the fractional value of its area, we can construct a ratio to solve for angle \(\displaystyle BOC\):

\(\displaystyle \frac{\Theta }{360}=\frac{1}{6}\)

\(\displaystyle \Theta=\frac{1}{6}*360^{\circ}=60^{\circ}\)

So, to find angle \(\displaystyle AOC\), we just need to subtract our other values from \(\displaystyle 360\):

\(\displaystyle \angle AOC= 360^{\circ}-\angle AOB - \angle BOC= 360^{\circ}-180^{\circ}-60^{\circ}=120^{\circ}\)

So, \(\displaystyle \angle AOC=120^{\circ}\).

Example Question #1 : Calculating The Angle Of A Sector

The radius of Circle A is equal to the perimeter of Square B. A sector of Circle A has the same area as Square B. Which of the following is the degree measure of this sector?

Possible Answers:

\(\displaystyle \left (\frac{45}{2\pi} \right )^{\circ }\)

\(\displaystyle 22 \frac{1}{2}^{\circ }\)

\(\displaystyle 45^{\circ }\)

\(\displaystyle \left (\frac{90}{\pi} \right )^{\circ }\)

\(\displaystyle \left (\frac{45}{ \pi} \right )^{\circ }\)

Correct answer:

\(\displaystyle \left (\frac{45}{2\pi} \right )^{\circ }\)

Explanation:

Call the length of a side of Square B \(\displaystyle r\). Its perimeter is \(\displaystyle 4r\), which is the radius of Circle A.

The area of the circle is \(\displaystyle A_{a}= \pi \left ( 4r\right )^{2} = 16 \pi r^{2}\); that of the square is \(\displaystyle A_{b}= r^{2}\). Therefore, a sector of the circle with area \(\displaystyle r^{2}\) will be \(\displaystyle \frac{1}{16 \pi}\) of the circle, which is a sector of measure

\(\displaystyle \frac{1}{16 \pi} \times 360 ^{\circ } = \left (\frac{45}{2\pi} \right )^{\circ }\)

Example Question #2 : Calculating The Angle Of A Sector

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Angle \(\displaystyle \widehat{DCB}\) is \(\displaystyle 35^{\circ}\). What is angle \(\displaystyle \widehat{DAB}\) ?

Possible Answers:

\(\displaystyle 55^{\circ}\)

\(\displaystyle 65^{\circ}\)

\(\displaystyle 75^{\circ}\)

\(\displaystyle 60^{\circ}\)

\(\displaystyle 70^{\circ}\)

Correct answer:

\(\displaystyle 70^{\circ}\)

Explanation:

This is the kind of question we can't get right if we don't know the trick. In a circle, the size of an angle at the center of the circle, formed by two segments intercepting an arc, is twice the size of the angle formed by two lines intercepting the same arc, provided one of these lines is the diameter of the circle. in other words, \(\displaystyle \widehat{DAB}\) is twice \(\displaystyle \widehat{DCB}\).

Thus,

\(\displaystyle 2(35^\circ)=70^\circ\)

Example Question #261 : Gmat Quantitative Reasoning

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\(\displaystyle A,B,C,D,E\) are \(\displaystyle 5\) evenly spaced points on the circle. What is angle \(\displaystyle \widehat{DFC}\)?

Possible Answers:

\(\displaystyle 80^{\circ}\)

\(\displaystyle 78^{\circ}\)

\(\displaystyle 144^{\circ}\)

\(\displaystyle 72^{\circ}\)

\(\displaystyle 36^{\circ}\)

Correct answer:

\(\displaystyle 72^{\circ}\)

Explanation:

We can see that the points devide the \(\displaystyle 360^{\circ}\) of the circle in 5 equal portions.

The final answer is given simply by \(\displaystyle \frac{360}{5}\) which is \(\displaystyle 72^{\circ}\), this is the angle of a slice of a pizza cut in 5 parts if you will!

Example Question #264 : Problem Solving Questions

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The \(\displaystyle 6\) points \(\displaystyle A,B,C,D,E\) and \(\displaystyle F\) are evenly spaced on the circle of center \(\displaystyle G\). What is the size of angle \(\displaystyle \widehat{FBE}\)?

Possible Answers:

\(\displaystyle 60^{\circ}\)

\(\displaystyle 32^{\circ}\)

\(\displaystyle 45^{\circ}\)

\(\displaystyle 35^{\circ}\)

\(\displaystyle 30^{\circ}\)

Correct answer:

\(\displaystyle 30^{\circ}\)

Explanation:

As we have seen previously, the 6 points divide the \(\displaystyle 360^{\circ}\) of the circle in 6 portion of same angle. Each portion form an angle of \(\displaystyle \frac{360}{6}\) or 60 degrees. As we also have previously seen, the angle formed by the lines intercepting an arc is twice more at the center of the circle than at the intersection of the lines intercepting the same arc with the circle, provided one of these lines is the diameter. In other words, \(\displaystyle \widehat{FGE}=2\widehat{FBE}\). Since \(\displaystyle \widehat{FGE}\) is 60 degrees, than, \(\displaystyle \widehat{FBE}\) must be 30 degrees, this is our final answer.  

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