Question

10 points

Answer 1

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;

  $4x+2=7x-19$

$-3x+2=-19$

     $-3x=-21$

         $x=7$

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 = $4(7)+2=28+2=30$

Length of QR = $7(7)-19=49-19=30$

Radius of the circle:

The radius of the circle PU can be determined using the Pythagorean theorem.

Thus, we have;

$QU^2=PQ^2+PU^2$

Substituting the values, we get;

 $34^2=30^2+PU^2$

$1156=900+PU^2$

 $256=PU^2$

   $16=PU$

Thus, the radius of the circle is 16.

Length of ST:

The length of ST is $ST=SU+UT$

Substituting SU = UT = 16 because the radius of the circle is 16.

Thus, we have;

$ST=16+16$

$ST=32$

Thus, the length of ST is 32.