A circle with center O has a radius of 15. Chord AB of the circle has a length of 18, and … — SAT

A circle with center $O$ has a radius of 15. Chord $AB$ of the circle has a length of 18, and chord $CD$ is parallel to $AB$ and has a length of 24. If the two chords are on opposite sides of the center $O$ , what is the distance between chord $AB$ and chord $CD$ ?

Cevap: 21

Cevap

The distance between the chords is 21. By drawing a line through the center

O

perpendicular to both chords, we bisect chord

AB

M

(length 9) and chord

CD

N

(length 12). Using the Pythagorean theorem with the radius of 15, the perpendicular distance from the center to chord

AB

\sqrt{15^2 - 9^2} = 12

and to chord

CD

\sqrt{15^2 - 12^2} = 9

. Since the chords lie on opposite sides of the center, the total distance between them is the sum of these distances,

12 + 9 = 21

Adım Adım Çözüm

Find the perpendicular distance from the center

O

to chord

AB

OM = 12

The perpendicular from the center bisects the chord, so we use the Pythagorean theorem in right triangle

OMA

with hypotenuse

OA = 15

and leg

AM = 9

Find the perpendicular distance from the center

O

to chord

CD

ON = 9

Similarly, we use the Pythagorean theorem in right triangle

ONC

with hypotenuse

OC = 15

and leg

CN = 12

Add the perpendicular distances together to find the total distance between the chords.

Because the chords are parallel and on opposite sides of the center, the total distance between them is the sum of their individual distances to the center.

Anahtar Kavram

The perpendicular from the center of a circle to a chord bisects the chord, and the distance from the center to the chord can be calculated using the Pythagorean theorem with the radius of the circle.

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