Common Core: High School - Geometry : Circles

Study concepts, example questions & explanations for Common Core: High School - Geometry

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All Common Core: High School - Geometry Resources

6 Diagnostic Tests 114 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #21 : Circles

What is the measure of an inscribed angle with an arc measurement of \(\displaystyle 86 ^{\circ}\)?

Possible Answers:

\(\displaystyle 43\degree\)

\(\displaystyle 172\degree\)

\(\displaystyle 86\degree\)

\(\displaystyle 344\degree\)

\(\displaystyle 87\degree\)

Correct answer:

\(\displaystyle 43\degree\)

Explanation:

The inscribed angle is simply half the arc measurement.

\(\displaystyle \\=86 \cdot \frac{1}{2} \\\\=43\)

Example Question #22 : Circles

What is the measure of an inscribed angle with an arc measurement of \(\displaystyle 54 ^{\circ}\)?

Possible Answers:

\(\displaystyle 216\degree\)

\(\displaystyle 55\degree\)

\(\displaystyle 27\degree\)

\(\displaystyle 54\degree\)

\(\displaystyle 108\degree\)

Correct answer:

\(\displaystyle 27\degree\)

Explanation:

The inscribed angle is simply half the arc measurement.

\(\displaystyle \\=54 \cdot \frac{1}{2} \\\\=27\)

Example Question #23 : Circles

What is the measure of an inscribed angle with an arc measurement of \(\displaystyle 13 ^{\circ}\)?

Possible Answers:

\(\displaystyle \frac{13}{2}\degree\)

\(\displaystyle 26\degree\)

\(\displaystyle 52\degree\)

\(\displaystyle 14\degree\)

\(\displaystyle 13\degree\)

Correct answer:

\(\displaystyle \frac{13}{2}\degree\)

Explanation:

The inscribed angle is simply half the arc measurement.

\(\displaystyle \\=13 \cdot \frac{1}{2} \\\\=\frac{13}{2}\)

Example Question #24 : Circles

What is the measure of an inscribed angle with an arc measurement of \(\displaystyle 31 ^{\circ}\)?

Possible Answers:

\(\displaystyle 32\degree\)

\(\displaystyle 62\degree\)

\(\displaystyle 31\degree\)

\(\displaystyle \frac{31}{2}\degree\)

\(\displaystyle 124\degree\)

Correct answer:

\(\displaystyle \frac{31}{2}\degree\)

Explanation:

The inscribed angle is simply half the arc measurement.

\(\displaystyle \\=31 \cdot \frac{1}{2} \\\\=\frac{31}{2}\)

Example Question #25 : Circles


From the following picture, determine \(\displaystyle x\), and \(\displaystyle y\).


Plot2

 

Possible Answers:

\(\displaystyle y = 111.0 , x = 100.0\)

\(\displaystyle y = 280.0 , x = 291.0\)

\(\displaystyle y = 100.0 , x = 111.0\)

\(\displaystyle y = 80.0 , x = 69.0\)

\(\displaystyle y = 260.0 , x = 249.0\)

Correct answer:

\(\displaystyle y = 100.0 , x = 111.0\)

Explanation:

Since this polygon is inscribed within a circle, we know a few things.

Plot2

The first thing we know is that the sum of all the interior angles must equal \(\displaystyle 360^{\circ}\).

The last thing we know, the most important one is all opposite angles must equal \(\displaystyle 180^{\circ}\).

Now we need to set up equations to solve for \(\displaystyle x\), and \(\displaystyle y\).

\(\displaystyle \\180 = y + 80.0 \\180 = x + 69.0\)

Now let's solve for \(\displaystyle x\), and \(\displaystyle y\).

\(\displaystyle \\180 -80.0= y + 80.0{\color{Red} -80.0} \\180-69.0 = x + 69.0{\color{Red} -69.0}\)

\(\displaystyle \\y = 100.0 \\x = 111.0\)

Example Question #2 : Inscribed And Circumscribed Circle Of Triangles: Ccss.Math.Content.Hsg C.A.3

From the following picture, determine \(\displaystyle x\) and \(\displaystyle y\).


Plot3

 

Possible Answers:

\(\displaystyle y = 268.0 , x = 311.0\)

\(\displaystyle y = 92.0 , x = 49.0\)

\(\displaystyle y = 131.0 , x = 88.0\)

\(\displaystyle y = 272.0 , x = 229.0\)

\(\displaystyle y = 88.0 , x = 131.0\)

Correct answer:

\(\displaystyle y = 88.0 , x = 131.0\)

Explanation:

Since this polygon is inscribed within a circle, we know a few things.

Plot3

The first thing we know is that the sum of all the interior angles must equal \(\displaystyle 360^{\circ}\).

The last thing we know, the most important one is all opposite angles must equal \(\displaystyle 180^{\circ}\).

Now we need to set up equations to solve for  \(\displaystyle x\) and \(\displaystyle y\).

\(\displaystyle \\180 = y + 92.0 \\180 = x + 49.0\)

Now let's solve for  \(\displaystyle x\) and \(\displaystyle y\).

\(\displaystyle \\180-92.0 = y + 92.0{\color{Red} -92.0} \\180-49.0 = x + 49.0{\color{Red} -49.0}\)

\(\displaystyle \\y = 88.0 \\x = 131.0\)

Example Question #25 : Circles

From the following picture, determine \(\displaystyle x\) and \(\displaystyle y\).


Plot4

Possible Answers:

\(\displaystyle y = 243.0 , x = 274.0\)

\(\displaystyle y = 63.0 , x = 94.0\)

\(\displaystyle y = 117.0 , x = 86.0\)

\(\displaystyle y = 94.0 , x = 63.0\)

\(\displaystyle y = 297.0 , x = 266.0\)

Correct answer:

\(\displaystyle y = 63.0 , x = 94.0\)

Explanation:

Since this polygon is inscribed within a circle, we know a few things.

Plot4

The first thing we know is that the sum of all the interior angles must equal \(\displaystyle 360^{\circ}\).

The last thing we know, the most important one is all opposite angles must equal \(\displaystyle 180^{\circ}\).

Now we need to set up equations to solve for  \(\displaystyle x\) and \(\displaystyle y\).

\(\displaystyle \\180 = y + 117.0 \\180 = x + 86.0\)

Now let's solve for \(\displaystyle x\) and \(\displaystyle y\).

\(\displaystyle \\180-117.0 = y + 117.0{\color{Red} -117.0} \\180-86.0 = x + 86.0{\color{Red} -86.0}\)

\(\displaystyle \\y = 63.0 \\x = 94.0\)

Example Question #26 : Circles

From the following picture, determine \(\displaystyle x\) and \(\displaystyle y\).


Plot5

Possible Answers:

\(\displaystyle y = 85.0 , x = 108.0\)

\(\displaystyle y = 108.0 , x = 85.0\)

\(\displaystyle y = 288.0 , x = 265.0\)

\(\displaystyle y = 252.0 , x = 275.0\)

\(\displaystyle y = 72.0 , x = 95.0\)

Correct answer:

\(\displaystyle y = 108.0 , x = 85.0\)

Explanation:

Since this polygon is inscribed within a circle, we know a few things.

Plot5

The first thing we know is that the sum of all the interior angles must equal \(\displaystyle 360^{\circ}\).

The last thing we know, the most important one is all opposite angles must equal \(\displaystyle 180^{\circ}\).

Now we need to set up equations to solve for \(\displaystyle x\) and \(\displaystyle y\).

\(\displaystyle \\180 = y + 72.0 \\180 = x + 95.0\)

Now let's solve for \(\displaystyle x\) and \(\displaystyle y\).

\(\displaystyle \\180 -72.0= y + 72.0{\color{Red} -72.0} \\180 -95.0= x + 95.0{\color{Red} -95.0}\)

\(\displaystyle \\y = 108.0 \\x = 85.0\)

 

 

Example Question #27 : Circles

From the following picture, determine \(\displaystyle x\) and \(\displaystyle y\).


Plot6

Possible Answers:

\(\displaystyle y = 291.0 , x = 288.0\)

\(\displaystyle y = 111.0 , x = 108.0\)

\(\displaystyle y = 69.0 , x = 72.0\)

\(\displaystyle y = 108.0 , x = 111.0\)

\(\displaystyle y = 249.0 , x = 252.0\)

Correct answer:

\(\displaystyle y = 111.0 , x = 108.0\)

Explanation:

Since this polygon is inscribed within a circle, we know a few things.

Plot6

The first thing we know is that the sum of all the interior angles must equal \(\displaystyle 360^{\circ}\).

The last thing we know, the most important one is all opposite angles must equal \(\displaystyle 180^{\circ}\).

Now we need to set up equations to solve for \(\displaystyle x\) and \(\displaystyle y\).

\(\displaystyle \\180 = y + 69.0 \\180 = x + 72.0\)

Now let's solve for  \(\displaystyle x\) and \(\displaystyle y\).

\(\displaystyle \\180-69.0 = y + 69.0{\color{Red} -69.0} \\180-72.0 = x + 72.0{\color{Red} -72.0} \\y = 111.0 \\x = 108.0\)

 

Example Question #28 : Circles

From the following picture, determine \(\displaystyle x\), and \(\displaystyle y\).

Plot7

Possible Answers:

\(\displaystyle y = 106.0 , x = 108.0\)

\(\displaystyle y = 254.0 , x = 252.0\)

\(\displaystyle y = 108.0 , x = 106.0\)

\(\displaystyle y = 74.0 , x = 72.0\)

\(\displaystyle y = 286.0 , x = 288.0\)

Correct answer:

\(\displaystyle y = 106.0 , x = 108.0\)

Explanation:

Since this polygon is inscribed within a circle, we know a few things.

Plot7

The first thing we know is that the sum of all the interior angles must equal \(\displaystyle 360^{\circ}\).

The last thing we know, the most important one is all opposite angles must equal \(\displaystyle 180^{\circ}\).

Now we need to set up equations to solve for \(\displaystyle x\), and \(\displaystyle y\).

\(\displaystyle \\180 = y + 74.0 \\180 = x + 72.0\)

Now let's solve for \(\displaystyle x\), and \(\displaystyle y\).

\(\displaystyle \\180 -74.0= y + 74.0{\color{Red} -74.0} \\180 -72.0= x + 72.0{\color{Red} -72.0} \\y = 106.0 \\x = 108.0\)

All Common Core: High School - Geometry Resources

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