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Intermediate Geometry : Circles

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Example Question #151 : Circles

Find the length of the chord in the figure below.

Possible Answers:

7.55

16.21

9.03

15.10

Correct answer:

15.10

Explanation:

When a chord is intercepted by a perpendicular line segment originating at the center of the circle, the chord is bisected, or cut in half.

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{11^2-8^2}=\sqrt{57}$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=2\sqrt{57}=15.10$

Make sure to round to places after the decimal.

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Example Question #151 : Circles

Find the length of the chord in the figure below.

Possible Answers:

3.31

7.62

5.19

6.63

Correct answer:

6.63

Explanation:

When a chord is intercepted by a perpendicular line segment originating at the center of the circle, the chord is bisected, or cut in half.

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{6^2-5^2}=\sqrt{11}$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=2\sqrt{11}=6.63$

Make sure to round to places after the decimal.

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Example Question #21 : Chords

Find the area of the shaded region in the figure below.

Possible Answers:

718.23

703.54

800.06

698.48

Correct answer:

703.54

Explanation:

When a chord is intercepted by a perpendicular line segment originating at the center of the circle, the chord is bisected, or cut in half. From the figure, you should notice that the base of the triangle is also the chord of the circle.

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{16^2-7^2}=\sqrt{207}$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=\text{Base of Triangle}=2\sqrt{207}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(2\sqrt{207}\times 7)=7\sqrt{207}$

Next, find the area of the circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 16^2=256\pi$

Finally, find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=256\pi-7\sqrt{207}=703.54$

Make sure to round to places after the decimal.

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Example Question #22 : Chords

Find the area of the shaded region in the figure below.

Possible Answers:

1709.06

1590.02

1663.50

1689.51

Correct answer:

1663.50

Explanation:

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{25^2-15^2}=\sqrt{400}=20$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=\text{Base of Triangle}=2(20)=40$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(40\times 15)=300$

Next, find the area of the circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 25^2=625\pi$

Finally, find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=625\pi-300=1663.50$

Make sure to round to places after the decimal.

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Example Question #23 : Chords

Find the area of the shaded region in the figure below.

Possible Answers:

970.16

590.07

889.74

981.16

Correct answer:

970.16

Explanation:

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{19^2-16^2}=\sqrt{105}$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=\text{Base of Triangle}=2\sqrt{105}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(2\sqrt{105}\times 16)=16\sqrt{105}$

Next, find the area of the circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 19^2=361\pi$

Finally, find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=361\pi-16\sqrt{105}=970.16$

Make sure to round to places after the decimal.

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Example Question #24 : Chords

Find the area of the shaded region in the figure below.

Possible Answers:

2098.98

2115.69

2071.51

2081.96

Correct answer:

2081.96

Explanation:

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{28^2-22^2}=\sqrt{300}$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=\text{Base of Triangle}=2\sqrt{300}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(2\sqrt{300}\times 22)=22\sqrt{300}$

Next, find the area of the circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 28^2=784\pi$

Finally, find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=784\pi-22\sqrt{300}=2081.96$

Make sure to round to places after the decimal.

Report an Error

Example Question #25 : Chords

Find the length of chord .

Possible Answers:

8.36

10.51

7.99

9.17

Correct answer:

9.17

Explanation:

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

$\text{Radius}^2=\text{distance}^2+\text{half chord}^2$

Rearrange the equation to solve for the length of half the chord.

$\text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}$

Now, multiply this value by two to find the length of the entire chord.

$\text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}$

Plug in the given radius and distance to find the length of the chord.

$\text{Chord}=2\sqrt{5^2-2^2}=2\sqrt{21}=9.17$

Make sure to round to two places after the decimal.

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Example Question #26 : Chords

Find the length of chord .

Possible Answers:

11.92

12.49

13.12

10.05

Correct answer:

12.49

Explanation:

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

$\text{Radius}^2=\text{distance}^2+\text{half chord}^2$

Rearrange the equation to solve for the length of half the chord.

$\text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}$

Now, multiply this value by two to find the length of the entire chord.

$\text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}$

Plug in the given radius and distance to find the length of the chord.

$\text{Chord}=2\sqrt{8^2-5^2}=2\sqrt{39}=12.49$

Make sure to round to two places after the decimal.

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Example Question #27 : Chords

Find the length of chord .

Possible Answers:

20.09

23.24

29.51

25.12

Correct answer:

23.24

Explanation:

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

$\text{Radius}^2=\text{distance}^2+\text{half chord}^2$

Rearrange the equation to solve for the length of half the chord.

$\text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}$

Now, multiply this value by two to find the length of the entire chord.

$\text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}$

Plug in the given radius and distance to find the length of the chord.

$\text{Chord}=2\sqrt{12^2-3^2}=2\sqrt{135}=23.24$

Make sure to round to two places after the decimal.

Report an Error

Example Question #28 : Chords

Find the length of the chord .

Possible Answers:

44.29

49.05

41.57

42.86

Correct answer:

41.57

Explanation:

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

$\text{Radius}^2=\text{distance}^2+\text{half chord}^2$

Rearrange the equation to solve for the length of half the chord.

$\text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}$

Now, multiply this value by two to find the length of the entire chord.

$\text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}$

Plug in the given radius and distance to find the length of the chord.

$\text{Chord}=2\sqrt{24^2-12^2}=2\sqrt{432}=41.57$

Make sure to round to two places after the decimal.

Report an Error

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Intermediate Geometry : Circles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #151 : Circles

Example Question #151 : Circles

Example Question #21 : Chords

Example Question #22 : Chords

Example Question #23 : Chords

Example Question #24 : Chords

Example Question #25 : Chords

Example Question #26 : Chords

Example Question #27 : Chords

Example Question #28 : Chords

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