Question

In circle M, secant line SQ and secant line SU intersect at point S outside the circle.
Segment SR = 4, segment SQ = 10, and segment ST = 2 . What is the length of segment TU?
5
12
18
20

Answer 1

To find the length of segment \( TU \), we can use the Power of a Point theorem. According to the theorem, if a point \( S \) is outside a circle and secants \( SQ \) and \( SU \) intersect the circle, then the following relationship holds:

\[
SR \cdot SQ = ST \cdot SU
\]

Here, we know:
- \( SR = 4 \)
- \( SQ = 10 \)
- \( ST = 2 \)
- We need to find \( TU \) so that \( SU = ST + TU \).

Thus, we can express \( SU \) as:
\[
SU = ST + TU = 2 + TU
\]

Now, substituting the known values into the Power of a Point formula:
\[
SR \cdot SQ = ST \cdot SU
\]
\[
4 \cdot 10 = 2 \cdot (2 + TU)
\]
Calculating the left side:
\[
40 = 2(2 + TU)
\]

Next, we expand the right side:
\[
40 = 4 + 2TU
\]

Now, we will isolate \( TU \):
\[
40 - 4 = 2TU
\]
\[
36 = 2TU
\]
Dividing both sides by 2 gives:
\[
TU = 18
\]

Thus, the length of segment \( TU \) is \( \boxed{18} \).