Question

what is rs?
m n o
14
7
p
l
31
9
r
q
s
rs =
submit

Explanation:

Step1: Use the property of a tangential quadrilateral

In a tangential quadrilateral (a quadrilateral with an incircle), the sum of the lengths of two opposite sides is equal to the sum of the lengths of the other two opposite sides. That is \(ML + OP+RS = MN + OQ+SL\). Also, we know that the lengths of the two - tangent segments from a single external point to a circle are equal. So \(ML = MS\), \(OP = OQ\), \(RS = RL\), \(NL=NM\).

The formula for a tangential quadrilateral is \(MS + OQ+RS=MN + OP+SL\). Since \(MS = 31\), \(MN = 14\), \(OQ=7 + 9=16\) (because \(OP = 7\) and \(PQ = 9\) and \(OQ=OP + PQ\))

We use the property \(MS+RQ=MN + SQ\) (where \(SQ\) is composed of segments related to the tangents). The property of the tangential quadrilateral \(a + c=b + d\) (where \(a\) and \(c\) are opposite sides, \(b\) and \(d\) are opposite sides). Here \(MS+PQ=MN + RS\)

Step2: Substitute the values into the formula

Substitute \(MS = 31\), \(MN = 14\), \(PQ = 9\) into the formula \(MS+PQ=MN + RS\)

We get \(31+9=14 + RS\)

\(RS=31 + 9-14\)

\(RS = 26\)

Answer:

\(26\)