Question

A
D
d
a
C
E
C
b
B
Qingyun has constructed a circle with center O
and chords AB
and CD
that intersect at point E
inside the circle. According to her measurements, segment a=13
mm, segment b=20
mm, and segment d=10
mm. What is the length of chord CD
?

(1 point)
Responses

Chord CD
is 33 mm.
Chord upper C upper D is 33 mm.

Chord CD
is 36 mm.
Chord upper C upper D is 36 mm.

Chord CD
is 26 mm.
Chord upper C upper D is 26 mm.

Chord CD
is 260 mm.
Chord upper C upper D is 260 mm.

Answer 1

To find the length of chord \( CD \), we can use the intersecting chords theorem. According to this theorem, if two chords intersect inside a circle, then the products of the lengths of the segments of each chord are equal.

In this case, we have:
- Chord \( AB \) with segments \( AE = a = 13 \) mm and \( EB = b = 20 \) mm.
- Chord \( CD \) with segments \( CE = c \) and \( ED = d = 10 \) mm.

Using the theorem, we have:
\[ AE \cdot EB = CE \cdot ED \]

Substituting the known values:
\[ 13 \cdot 20 = CE \cdot 10 \]

Calculating \( 13 \cdot 20 \):
\[ 260 = CE \cdot 10 \]

Now, solve for \( CE \):
\[ CE = \frac{260}{10} = 26 \text{ mm} \]

Now, since \( CD = CE + ED \), we can find the length of chord \( CD \):
\[ CD = CE + ED = 26 + 10 = 36 \text{ mm} \]

Hence, the length of chord \( CD \) is \(\boxed{36 \text{ mm}}\).