Question

1. find the perimeter of quadrilateral pqrs.

Explanation:

Step1: Use the property of tangents to a circle from a common - point

If a quadrilateral \(PQRS\) has an incircle (a circle tangent to all four sides), then 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 \(PQ + SR=PS + QR\).
Let \(PQ = 37.5\), \(PS = 23\), \(QR = 27.1\), \(SR = 52.5\).

Step2: Calculate the perimeter \(P\) of the quadrilateral

The perimeter \(P\) of a quadrilateral \(PQRS\) is \(P=PQ + QR+RS + SP\).
Since \(PQ + SR=PS + QR\), we can also calculate the perimeter as \(P = 2(PQ + SR)\) (or \(2(PS + QR)\)).
Substitute \(PQ = 37.5\) and \(SR = 52.5\) into the formula:
\(P=2(37.5 + 52.5)\)

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Answer:

The perimeter of quadrilateral \(PQRS\) is \(180\).