Question

In circle M
, secant line SQ
and secant line SU
intersect at point S
outside the circle. If segment SQ=21
, segment SR=8
, and segment ST=4
, find the length of segment SU
.

(1 point)

Answer 1

To solve for the length of segment \( SU \), we can use the secant-secant power theorem. This theorem states that if two secant lines intersect at an external point, the products of the segments of each secant line are equal.

In this case, we have:

- For secant line \( SQ \):
\[ SQ \times SR = 21 \times 8 = 168 \]

- For secant line \( SU \) (where \( ST \) is a segment of \( SU \), with \( ST = 4 \)):
Let \( SU = x \) (the total length of segment \( SU \), which we are trying to find).
The relationship can be expressed as:
\[ SU \times ST = x \times 4 \]

According to the secant-secant theorem, we set these equal to each other:
\[ SQ \times SR = SU \times ST \]
\[ 168 = x \times 4 \]

Now, we can solve for \( x \):
\[ x = \frac{168}{4} = 42 \]

Thus, the length of segment \( SU \) is \( \boxed{42} \).