Please help me I’m so lost

Length: About 7.61 millimeters

Step-by-step explanation:

Let's first calculate the circumference of Circle W. Now given that the radius WS is 4.5 millimeters, apply the circumference formula as such to get the circumference:

Circumference = 2πr = 2π * (4.5) = 2 * (4.5)π = 9π

The circumference will come in handy later, so now let us calculate the degree measure of arc TS. We can see that TQ and PR act as diameters. From this we conclude that:

mPT + mPQ = 180,

mPT = mQR (Vertical, thus congruent),

mQR + mRS + mTS = 180

Let's substitute known values into the first equation:

mPT + mPQ = 180,

mPT + 128 = 180,

mPT = 52 degrees

This means that:

mQR = 52 degrees

Now let's substitute known values into the third equation:

mQR + mRS + mTS = 180,

52 + 31 + mTS = 180,

83 + mTS = 180,

mTS = 97 degrees

Knowing the degree measure of the arc we need to calculate, we can apply a proportionality including the circumference as such:

mTS / 360 = TS / Circumference

Let's substitute known values into the equation and solve for TS:

97 / 360 = TS / 9π,

97 (9π) = 360 (TS)    (Cross multiplication),

873π = 360 (TS),

TS = 873π / 360

TS = 2.425π

TS = 2.425 (3.14159265)

TS = About 7.62 or 7.61 (depends on length of π)

55 degrees because 55+55=110 degrees