How to find the length of a chord - Intermediate Geometry

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Question

The radius of $\small \odot O$ is feet and $\small m\widehat{AB}=60^{\circ}$ . Find the length of chord $\small \overline{AB}$ .

Answer

We begin by drawing in three radii: one to $\small A$ , one to $\small B$ , and one perpendicular to $\small \overline{AB}$ with endpoint $\small C$ on our circle.

We must also recall that our central angle $\small \angle AOB$ has a measure equal to its intercepted arc. Therefore, $\small m\angle AOB=60^\circ$ . Our perpendicular radius actually divides $\small \triangle AOB$ into two congruent triangles. Therefore, it also bisects our central angle, meaning that $\small m\angle AOC=m\angle COB=30^\circ$

Therefore, each of these triangles is a 30-60-90 triangle, meaning that each half of our chord is simply half the length of the hypotenuse (our radius which is 6). Therefore, each half is 3, and the entire chord is 6 feet.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$\rightarrow \sqrt{24}=4.8989$

Since this leg is half of the chord, the total chord length is 2 times that, or 9.798.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$10^2-6^2=64 \rightarrow \sqrt{64}=8$

Since this leg is half of the chord, the total chord length is 2 times that, or 16.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$6^2-4.5^2=15.75 \rightarrow \sqrt{15.75}=3.9686$

Since this leg is half of the chord, the total chord length is 2 times that, or 7.937.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$3.5^2-3^2=3.25 \rightarrow \sqrt{3.25}=1.8027$

Since this leg is half of the chord, the total chord length is 2 times that, or 3.606.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$5^2-4^2=9 \rightarrow \sqrt{9}=3$

Since this leg is half of the chord, the total chord length is 2 times that, or 6.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$12^2-10^2=44 \rightarrow \sqrt{44}=6.6332$

Since this leg is half of the chord, the total chord length is 2 times that, or 13.266.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$3^2-2^2=5 \rightarrow \sqrt{5}=2.2361$

Since this leg is half of the chord, the total chord length is 2 times that, or 4.472.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$4^2-1^2=15 \rightarrow \sqrt{15}=3.873$

Since this leg is half of the chord, the total chord length is 2 times that, or 7.746.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$12^2-11^2=23 \rightarrow \sqrt{23}=4.7958$

Since this leg is half of the chord, the total chord length is 2 times that, or 9.592.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$10^2-9^2=19 \rightarrow \sqrt{19}=4.3589$

Since this leg is half of the chord, the total chord length is 2 times that, or 8.718.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$4^2-2^2=12 \rightarrow \sqrt{12}=3.4641$

Since this leg is half of the chord, the total chord length is 2 times that, or 6.928.

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Question

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Answer

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$9^2-6^2=45 \rightarrow \sqrt{45}=6.7082$

Since this leg is half of the chord, the total chord length is 2 times that, or 13.416.

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Question

Find the measure of AB in the below circle.

Answer

When two chords intersect inside of a circle, one can find the length of the chord for this particular problem by solving the below equation.

Substitute the given information from the diagram.

$\(x)(40)=(25)(16) \40x=400 \x=10$

Now we add 10 to 40 to find the total length from A to B.

AB=50.

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Question

Find the length of the chord in the figure below.

Answer

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.

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

$\text{Half Chord}=\sqrt{14^2-12^2}=\sqrt{52}$

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

$\text{Length of chord}=2\sqrt{52}=14.42$

Make sure to round to places after the decimal.

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Question

Find the length of the chord in the figure below.

Answer

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.

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

$\text{Half Chord}=\sqrt{6^2-3^2}=\sqrt{27}$

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

$\text{Length of chord}=2\sqrt{27}=10.39$

Make sure to round to places after the decimal.

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Question

Find the length of the chord in the figure below.

Answer

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.

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

$\text{Half Chord}=\sqrt{9^2-8^2}=\sqrt{17}$

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

$\text{Length of chord}=2\sqrt{17}=8.25$

Make sure to round to places after the decimal.

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Question

Find the length of the chord in the figure below.

Answer

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.

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

$\text{Half Chord}=\sqrt{7^2-5^2}=\sqrt{24}$

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

$\text{Length of chord}=2\sqrt{24}=9.80$

Make sure to round to places after the decimal.

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Question

Find the length of the chord in the figure below.

Answer

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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Question

Find the length of the chord in the figure below.

Answer

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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