Chords AB and CD intersect at point E inside a circle. If AE = 6, EB = 8, and the total le… — ACT

Chords $AB$ and $CD$ intersect at point $E$ inside a circle. If $AE = 6$ , $EB = 8$ , and the total length of chord $CD$ is $16$ , what is the length of the shorter segment of chord $CD$ ?

Cevap: 4

Cevap

The length of the shorter segment of chord

CD

4

According to the Intersecting Chords Theorem, when two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other. For chords

AB

and

CD

intersecting at point

E

, this relationship is expressed as

AE \cdot EB = CE \cdot ED

. Substituting the given values yields

6 \cdot 8 = CE \cdot ED

, so

CE \cdot ED = 48

. Since the total length of chord

CD

16

, we can define

CE = x

and

ED = 16 - x

. The equation becomes

x(16 - x) = 48

, which simplifies to the quadratic equation

x^2 - 16x + 48 = 0

. Factoring this equation gives

(x - 12)(x - 4) = 0

, meaning the two segments of chord

CD

have lengths of

12

and

4

. The length of the shorter segment is

4

Adım Adım Çözüm

State the relationship between intersecting chord segments.

AE \cdot EB = CE \cdot ED

By the Intersecting Chords Theorem, the product of the segments of one chord equals the product of the segments of the other.

Substitute the known lengths and define the segments of

CD

using a variable

x

6 \cdot 8 = x(16 - x)

, which simplifies to

48 = 16x - x^2

We are given

AE = 6

and

EB = 8

. Since the total length of chord

CD

16

, if one segment is

x

, the remaining segment must be

16 - x

Solve the quadratic equation for

x

by factoring.

x^2 - 16x + 48 = 0 \implies (x - 12)(x - 4) = 0

, so

x = 12

x = 4

Rearranging the equation into standard quadratic form allows us to find the two possible segment lengths.

Identify the shorter segment length from the two solutions.

4

The two segment lengths are

12

and

4

. The problem asks for the shorter segment, which is

4

Anahtar Kavram

Intersecting Chords Theorem

Soru

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