The measure of AB is 62 cm.

We can use the properties of a cyclic quadrilateral to find the measure of AB.

A quadrilateral is cyclic if and only if the sum of the opposite angles is equal to 180 degrees.

In this problem, we know that ∠DAB = 90 degrees and the quadrilateral is touching the circle at points P, Q, R, and S, so it is a cyclic quadrilateral.

Given that CR = 23 cm and CB = 39 cm, we can use the property of opposite sides of a cyclic quadrilateral being equal to find the measure of AB.

AB = CD = CR + CB = 23 cm + 39 cm = 62 cm

So, the measure of AB is 62 cm.

Another way to check this is by applying Ptolemy's theorem,

which states that in a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides along with the product of the distance between the center of the circle and the point of intersection of diagonals (r) and 2.

Here, CR x AB + CB x AD = r x (CD + AB)

23 x AB + 39 x AD = 14(62 + AD)

Solving this equation we get AB = 62cm

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