Answer:
ST = 32
Explanation:
Given circle U with tangents PQ = 4x+2 and QR = 7x-19 and segment QU = 34, you want the length of diameter ST.
Value of x
The two tangents are the same length, so ...
QR = PQ
7x -19 = 4x +2
3x = 21 . . . . . . . . add 19-4x
x = 7 . . . . . . . . divide by 3
The length of tangent PQ is ...
PQ = 4x +2 = 4(7) +2 = 30
Right triangle
Triangle QPU is a right triangle with leg PQ = 30 and hypotenuse QU = 34. Then the length of radius PU is found using the Pythagorean theorem:
PU² + PQ² = QU²
PU² = QU² - PQ² = 34² -30² = 256
PU = √256 = 16 . . . . . radius of the circle
Diameter
The diameter ST of the circle is twice the length of the radius:
ST = 2·PU = 2·16
ST = 32
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Additional comment
You may recognize the right triangle as having sides that are the Pythagorean triple {8, 15, 17} multiplied by 2. The lengths 30 and 34 give you the clue.