Question

what is op?
28
m
n
o
20
t
s
12
r
15
p
q
op =
submit
work it out

Explanation:

Step1: Use the property of tangents to a circle in a quadrilateral

In a quadrilateral circumscribed about a circle (tangential quadrilateral), 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 \(SM + OQ=SO + MQ\). Also, for a tangential quadrilateral, if we consider the tangent - segment lengths. Let \(MT = MN=x\), \(NR = OP = y\), \(RQ=RS = z\), \(ST=SM = w\). We know that \(SM + OQ=SO + MQ\). Another property: if we have a tangential quadrilateral \(SMNO\) (with an incircle), the lengths of the tangents from a single external point to a circle are equal. So \(SM=ST\), \(MN = NO\), \(OP=OQ\), \(RQ = RS\).
We know that \(SM = 20\) (since \(ST = 20\)), \(RQ=15\), \(RS = 12\).
We use the formula for the sum of opposite sides of a tangential quadrilateral: \(SM+OQ=SO + MQ\). Also, we can use the fact that if we let the lengths of the tangents from each vertex to the points of tangency. Let's use the following: \(SO=28\), \(SM = 20\), \(RQ = 15\), \(RS=12\).
We know that \(SM+OQ=SO + RQ+RS\). Substitute the values: \(20 + OQ=28+12 + 15\) is wrong. The correct formula based on tangent - segment equality: \(SM+OQ=SO + RQ\) (because \(SM = ST\), \(OQ = OP\), \(SO=MN + NO\) (but using the property \(a + c=b + d\) where \(a,b,c,d\) are the side lengths of the tangential quadrilateral. Let \(a = SM\), \(b = SO\), \(c = OQ\), \(d = MQ\). The correct property is \(SM+OQ=SO + RQ\) (since \(SM = ST\), \(OQ = OP\), \(SO=MN + NO\) and \(MQ=MT+RQ\) and \(MT = MN\), \(NO = OP\)).
We know that \(SM = 20\), \(SO = 28\), \(RQ = 15\). Rearranging the formula \(SM+OP=SO+(RQ - RS + RS)\) (no, better use \(SM+OP=SO+(RQ)\) (incorrect). The correct formula: In a tangential quadrilateral \(SMNO\) (with in - circle), \(SM + OQ=SO+MQ\). But using the tangent - segment lengths: \(SM = 20\), \(RS = 12\), \(RQ = 15\), \(SO = 28\).
We use \(SM+OP=SO+(RQ)\) (derived from \(MT = MN\), \(NR = OP\), \(RQ = RS\), \(ST = SM\)).
\(OP=SO-(SM - RQ)\)

Step2: Calculate the value of \(OP\)

Substitute \(SM = 20\), \(SO = 28\), \(RQ = 15\) into the formula \(OP=SO-(SM - RQ)\)

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Answer:

\(13\)