Question

given the circle below with chords \\(\overline{mn}\\) and \\(\overline{op}\\). find the length of \\(\overline{mq}\\). round to the nearest tenth if necessary.
image of a circle with chords mn and op intersecting at q. segment pq is 24, qn is 18, oq is 31, and mq is unknown.

Explanation:

Step1: Recall the Chord Intersection Theorem

When two chords intersect in a circle, the products of the lengths of their segments are equal. So, if chords \( \overline{MN} \) and \( \overline{OP} \) intersect at \( Q \), then \( MQ \times QN = OQ \times QP \).

Step2: Assign Variables and Substitute Values

Let \( MQ = x \). We know \( QN = 18 \), \( OQ = 31 \), and \( QP = 24 \). Substituting into the theorem: \( x \times 18 = 31 \times 24 \).

Step3: Solve for \( x \)

First, calculate the right - hand side: \( 31\times24 = 744 \). Then, solve for \( x \): \( x=\frac{744}{18}=\frac{124}{3}\approx41.3 \) (rounded to the nearest tenth).

Answer:

\( \approx41.3 \)