Given:
The lengths PQ = 4x + 2, QR = 7x – 19, and QU = 34
We need to determine the value of ST.
Value of x:
By two tangent theorem, we have;
PQ = QR
Substituting the values, we get;
[tex]4x+2=7x-19[/tex]
[tex]-3x+2=-19[/tex]
[tex]-3x=-21[/tex]
[tex]x=7[/tex]
Thus, the value of x is 7.
Length of PQ and QR:
Substituting x = 7 in the expressions of PQ and QR, we get the lengths.
Length of PQ = [tex]4(7)+2=28+2=30[/tex]
Length of QR = [tex]7(7)-19=49-19=30[/tex]
Radius of the circle:
The radius of the circle PU can be determined using the Pythagorean theorem.
Thus, we have;
[tex]QU^2=PQ^2+PU^2[/tex]
Substituting the values, we get;
[tex]34^2=30^2+PU^2[/tex]
[tex]1156=900+PU^2[/tex]
[tex]256=PU^2[/tex]
[tex]16=PU[/tex]
Thus, the radius of the circle is 16.
Length of ST:
The length of ST is [tex]ST=SU+UT[/tex]
Substituting SU = UT = 16 because the radius of the circle is 16.
Thus, we have;
[tex]ST=16+16[/tex]
[tex]ST=32[/tex]
Thus, the length of ST is 32.
