Math / Geometry

QuestionIn the diagram below, a chord of length 18 centimeters is bisected by a line segment that starts at the center and is 6 centimeters from the center. What is the radius of the circle in centimeters? A. $3 \sqrt{13}$ B. $3 \sqrt{5}$ C. 10 D. 12 E. 15

Studdy Solution

STEP 1

1. The chord is bisected by a line segment from the center of the circle.
2. The length of the chord is $18$ centimeters.
3. The perpendicular distance from the center to the chord is $6$ centimeters.
4. The circle's radius needs to be determined.

STEP 2

1. Understand the relationship between the chord, radius, and the perpendicular bisector.
2. Use the Pythagorean theorem to find the radius.

STEP 3

Understand the relationship between the chord, radius, and the perpendicular bisector:
- When a line segment from the center of a circle bisects a chord, it is perpendicular to the chord. - The bisected chord forms two equal segments, each of length $\frac{18}{2} = 9$ centimeters.

STEP 4

Use the Pythagorean theorem to find the radius:
- Consider the right triangle formed by the radius, half of the chord, and the perpendicular distance from the center to the chord. - Let $r$ be the radius of the circle. - According to the Pythagorean theorem:
$r^2 = 9^2 + 6^2$

STEP 5

Calculate the radius:
$r^2 = 81 + 36$ $r^2 = 117$ $r = \sqrt{117}$ $r = 3 \sqrt{13}$
The radius of the circle is:
$\boxed{3 \sqrt{13}}$

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QuestionIn the diagram below, a chord of length 18 centimeters is bisected by a line segment that starts at the center and is 6 centimeters from the center. What is the radius of the circle in centimeters? A. $3 \sqrt{13}$ B. $3 \sqrt{5}$ C. 10 D. 12 E. 15

Studdy Solution

STEP 1

1. The chord is bisected by a line segment from the center of the circle.
2. The length of the chord is $18$ centimeters.
3. The perpendicular distance from the center to the chord is $6$ centimeters.
4. The circle's radius needs to be determined.

STEP 2

1. Understand the relationship between the chord, radius, and the perpendicular bisector.
2. Use the Pythagorean theorem to find the radius.

STEP 3

Understand the relationship between the chord, radius, and the perpendicular bisector:
- When a line segment from the center of a circle bisects a chord, it is perpendicular to the chord. - The bisected chord forms two equal segments, each of length $\frac{18}{2} = 9$ centimeters.

STEP 4

Use the Pythagorean theorem to find the radius:
- Consider the right triangle formed by the radius, half of the chord, and the perpendicular distance from the center to the chord. - Let $r$ be the radius of the circle. - According to the Pythagorean theorem:
$r^2 = 9^2 + 6^2$

STEP 5

Calculate the radius:
$r^2 = 81 + 36$ $r^2 = 117$ $r = \sqrt{117}$ $r = 3 \sqrt{13}$
The radius of the circle is:
$\boxed{3 \sqrt{13}}$

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QuestionIn the diagram below, a chord of length 18 centimeters is bisected by a line segment that starts at the center and is 6 centimeters from the center. What is the radius of the circle in centimeters? A. 3133 \sqrt{13}313​ B. 353 \sqrt{5}35​ C. 10 D. 12 E. 15

Studdy Solution

STEP 1

1. The chord is bisected by a line segment from the center of the circle.2. The length of the chord is 18 18 18 centimeters.3. The perpendicular distance from the center to the chord is 6 6 6 centimeters.4. The circle's radius needs to be determined.

STEP 2

1. Understand the relationship between the chord, radius, and the perpendicular bisector.2. Use the Pythagorean theorem to find the radius.

STEP 3

Understand the relationship between the chord, radius, and the perpendicular bisector:- When a line segment from the center of a circle bisects a chord, it is perpendicular to the chord. - The bisected chord forms two equal segments, each of length 182=9 \frac{18}{2} = 9 218​=9 centimeters.

STEP 4

Use the Pythagorean theorem to find the radius:- Consider the right triangle formed by the radius, half of the chord, and the perpendicular distance from the center to the chord. - Let r r r be the radius of the circle. - According to the Pythagorean theorem:r2=92+62 r^2 = 9^2 + 6^2 r2=92+62

STEP 5

Calculate the radius:r2=81+36 r^2 = 81 + 36 r2=81+36 r2=117 r^2 = 117 r2=117 r=117 r = \sqrt{117} r=117​ r=313 r = 3 \sqrt{13} r=313​ The radius of the circle is:313 \boxed{3 \sqrt{13}} 313​​

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QuestionIn the diagram below, a chord of length 18 centimeters is bisected by a line segment that starts at the center and is 6 centimeters from the center. What is the radius of the circle in centimeters? A. $3 \sqrt{13}$ B. $3 \sqrt{5}$ C. 10 D. 12 E. 15

1. The chord is bisected by a line segment from the center of the circle.
2. The length of the chord is $18$ centimeters.
3. The perpendicular distance from the center to the chord is $6$ centimeters.
4. The circle's radius needs to be determined.

1. Understand the relationship between the chord, radius, and the perpendicular bisector.
2. Use the Pythagorean theorem to find the radius.

Understand the relationship between the chord, radius, and the perpendicular bisector:
- When a line segment from the center of a circle bisects a chord, it is perpendicular to the chord. - The bisected chord forms two equal segments, each of length $\frac{18}{2} = 9$ centimeters.

Use the Pythagorean theorem to find the radius:
- Consider the right triangle formed by the radius, half of the chord, and the perpendicular distance from the center to the chord. - Let $r$ be the radius of the circle. - According to the Pythagorean theorem:
$r^2 = 9^2 + 6^2$

Calculate the radius:
$r^2 = 81 + 36$ $r^2 = 117$ $r = \sqrt{117}$ $r = 3 \sqrt{13}$
The radius of the circle is:
$\boxed{3 \sqrt{13}}$