Question

In the following figure chord MN and chord RS intersect at point D. If RD = 15, DS = 4, MD = 8, find DN by completing the following activity: 

Activity : 
\(\therefore\) MD \(\times\) DN = \(\boxed{\phantom{SD}}\) \(\times\) DS \(\dots\) (Theorem of internal division of chords)
\(\therefore\) \(\boxed{\phantom{8}}\) \(\times\) DN = 15 \(\times\) 4
\(\therefore\) DN = \(\frac{\boxed{\phantom{60}}}{8}\)
\(\therefore\) DN = \(\boxed{\phantom{7.5}}\)

Hint:

Remember the intersecting chords theorem as "part \(\times\) part = part \(\times\) part". Make sure you are multiplying the segments of the same chord together.

Answer 1

Step 1: Understanding the Concept:
When two chords of a circle intersect inside the circle, the product of the segments of one chord is equal to the product of the segments of the other chord. This is known as the theorem of intersecting chords or internal division of chords.

Step 2: Key Formula or Approach:
For chords MN and RS intersecting at D, the theorem states: \[ MD \times DN = RD \times DS \]

Step 3: Detailed Explanation:
Here is the completed activity with the blanks filled in.
\(\therefore\) MD \(\times\) DN = \(\boxed{\text{RD}}\) \(\times\) DS \(\dots\) (Theorem of internal division of chords)
Given: RD = 15, DS = 4, MD = 8.
\(\therefore\) \(\boxed{8}\) \(\times\) DN = 15 \(\times\) 4
\(\therefore\) 8 \(\times\) DN = 60
\(\therefore\) DN = \(\frac{\boxed{60}}{8}\)
\(\therefore\) DN = \(\boxed{7.5}\)

Step 4: Final Answer:
The length of DN is 7.5.