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Intermediate Geometry : Circles

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #21 : How To Find The Length Of A Chord

Find the length of chord .

Possible Answers:

17.82

14.29

Correct answer:

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.

$\text{Chord}=2\sqrt{10^2-6^2}=2\sqrt{64}=16$

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Example Question #30 : Chords

Find the length of chord .

Possible Answers:

22.47

16.12

17.50

14.21

Correct answer:

16.12

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.

$\text{Chord}=2\sqrt{9^2-4^2}=2\sqrt{65}=16.12$

Make sure to round to two places after the decimal.

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Example Question #31 : How To Find The Length Of A Chord

Find the length of chord .

Possible Answers:

34.71

28.51

33.22

29.33

Correct answer:

29.33

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.

$\text{Chord}=2\sqrt{24^2-19^2}=2\sqrt{215}=29.33$

Make sure to round to two places after the decimal.

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Example Question #32 : How To Find The Length Of A Chord

Find the length of the chord .

Possible Answers:

9.50

9.80

9.14

9.96

Correct answer:

9.80

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.

$\text{Chord}=2\sqrt{5^2-1^2}=2\sqrt{24}=9.80$

Make sure to round to two places after the decimal.

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Example Question #161 : Circles

Find the length of the chord .

Possible Answers:

31.59

36.28

32.66

39.05

Correct answer:

36.28

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.

$\text{Chord}=2\sqrt{27^2-20^2}=2\sqrt{329}=36.28$

Make sure to round to two places after the decimal.

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Example Question #34 : How To Find The Length Of A Chord

Find the length of the chord .

Possible Answers:

21.56

29.60

Correct answer:

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.

$\text{Chord}=2\sqrt{13^2-5^2}=2\sqrt{144}=24$

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Example Question #35 : How To Find The Length Of A Chord

Find the length of the chord .

Possible Answers:

22.63

21.44

19.85

24.09

Correct answer:

22.63

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.

$\text{Chord}=2\sqrt{12^2-4^2}=2\sqrt{128}=22.63$

Make sure to round to two places after the decimal.

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Example Question #36 : How To Find The Length Of A Chord

Chords 1

Figure NOT drawn to scale.

Refer to the above diagram.

True, false, or undetermined: AX = 6 .

Possible Answers:

False

Undetermined

True

Correct answer:

True

Explanation:

If two chords intersect inside a circle, they cut each other into segments such that the product of the lengths of the portions of one chord is equal to that of the lengths of the portions of the other chord. In other words,

$AX \cdot XB = CX \cdot XD$

Set XB= 18, CX = 12, XD= 9 , and solve for :

$AX \cdot 18 = 12\cdot 9$

$AX \cdot 18 = 108$

$AX \cdot 18\div 18 = 108 \div 18$

AX = 6 .

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Example Question #37 : How To Find The Length Of A Chord

A circle with a radius of 10 centimeters is shown below. What is the area, in square centimeters, of the shaded region of the circle?

Possible Answers:

$100\pi -36$

$100\pi -48$

$36\pi -48$

$36\pi$

Correct answer:

$100\pi -48$

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, you 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.

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

$\text{Half Chord}=\sqrt{10^2-6^2}=\sqrt{64}=8$

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

$\text{Length of chord}=\text{Base of Triangle}=16$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}bh$

$\text{Area of Triangle}=\frac{1}{2}(16)(6)=48$

Next, find the area of the circle.

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

$\text{Area of Circle}=\pi\times 10^2=100\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=100\pi-48=266.2 \text{ cm}^{2}$

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Intermediate Geometry : Circles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #21 : How To Find The Length Of A Chord

Example Question #30 : Chords

Example Question #31 : How To Find The Length Of A Chord

Example Question #32 : How To Find The Length Of A Chord

Example Question #161 : Circles

Example Question #34 : How To Find The Length Of A Chord

Example Question #35 : How To Find The Length Of A Chord

Example Question #36 : How To Find The Length Of A Chord

Example Question #37 : How To Find The Length Of A Chord

All Intermediate Geometry Resources

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