Question

S
T
R
.
M
U
Lines SQ and SU are secants. If SR¯¯¯¯¯¯¯ = 12, SQ¯¯¯¯¯¯¯ = 30, ST¯¯¯¯¯¯¯ = 9 and TU¯¯¯¯¯¯¯=5x−24
, find x.

(1 point)
Responses

360
360

31
31

40
40

11
11

Answer 1

To solve for \( x \) using the given information about the secants, we can apply the secant segment theorem (also known as the Power of a Point theorem). According to this theorem, if two secants intersect outside of a circle, the products of the lengths of the segments of each secant are equal.

In this case, we have the secants \( SQ \) and \( SU \). The theorem states that:

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

Given the lengths:
- \( SR = 12 \)
- \( SQ = 30 \)
- \( ST = 9 \)
- \( TU = 5x - 24 \)

Thus, we can express \( SU \) as:

\[
SU = ST + TU = 9 + (5x - 24) = 5x - 15
\]

Now we can substitute the known values into the equation from the secant segment theorem:

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

Substituting the known lengths:

\[
12 \cdot 30 = 9 \cdot (5x - 15)
\]

Calculating the left side:

\[
360 = 9(5x - 15)
\]

Now, distribute the \( 9 \) on the right side:

\[
360 = 45x - 135
\]

Next, we add \( 135 \) to both sides:

\[
360 + 135 = 45x
\]

\[
495 = 45x
\]

Now, we divide both sides by \( 45 \):

\[
x = \frac{495}{45} = 11
\]

Thus, the value of \( x \) is \( \boxed{11} \).