Qingyun has constructed a circle with center O

Question

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)

Answer 1

To find the length of chord CD in this situation, we can use the intersecting chords theorem. This theorem states that if two chords AB and CD intersect at point E inside a circle, then the products of the lengths of the segments of each chord are equal.

According to the information given:

• Chord AB has segments AE = a = 13 mm and EB = b = 20 mm.
• Chord CD has segments CE = c and ED = d = 10 mm.

Using the theorem, we can express the relationship as follows:

\[ AE \times EB = CE \times ED \]

Substituting the known values into this equation:

\[ 13 \times 20 = CE \times 10 \]

Calculating the left side:

\[ 260 = CE \times 10 \]

Now, we can find CE by isolating it:

\[ CE = \frac{260}{10} = 26 , \text{mm} \]

Now that we have both segments of chord CD (CE and ED), we can find the length of chord CD:

\[ CD = CE + ED = 26 , \text{mm} + 10 , \text{mm} = 36 , \text{mm} \]

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