Question

Lines SQ and SU are secants. If ¯¯¯¯¯¯¯¯SR =12, ¯¯¯¯¯¯¯¯SQ =30, ¯¯¯¯¯¯¯ST = 9 and ¯¯¯¯¯¯¯¯ TU=5x−24, find the value of x.
(1 point)
Responses

11
11

40
40

360
360

31

Answer 1

To solve for the value of \( x \) in the given problem, we can use the Secant Segment Theorem, which states that if two secants intersect outside a circle, the product of the entire secant length and the external segment length are equal.

In this case, we have:

1. For secant \( SQ \):
- Entire length \( SQ = 30 \)
- External segment \( SR = 12 \)

2. For secant \( SU \):
- Entire length \( SU = ST + TU = 9 + (5x - 24) = 5x - 15 \)
- External segment \( ST = 9 \)

According to the theorem:
\[
SR \cdot SQ = ST \cdot SU
\]
Substituting the values we have:
\[
12 \cdot 30 = 9 \cdot (5x - 15)
\]
Calculating the left side:
\[
360 = 9 \cdot (5x - 15)
\]
Now, simplify the right side:
\[
360 = 45x - 135
\]
Next, add 135 to both sides:
\[
360 + 135 = 45x
\]
\[
495 = 45x
\]
Now, divide both sides by 45:
\[
x = \frac{495}{45} = 11
\]

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