Circle M has secant lines QR and UT that intersect at point S outside the circle. Find x if the measure of secant segment QS = 15, the measure of RS = 8, the measure of ST = 3x-2 and the measure of SU = 12

Question

S
T
R
M
U
Q
(1 point)
Responses

x = 10
x = 10

x = 4
x = 4

x = 120
x = 120

x= 12

Answer 1

To find the value of \(x\) using the given secant segments and the Power of a Point Theorem, we will state the theorem first:

The Power of a Point Theorem states that if a point \(S\) is outside a circle and two secant segments \(QR\) and \(UT\) intersect the circle, then:

\[ QS \cdot RS = SU \cdot ST \]

Given:

• \(QS = 15\)
• \(RS = 8\)
• \(ST = 3x - 2\)
• \(SU = 12\)

We can set up the equation using the Power of a Point Theorem:

\[ 15 \cdot 8 = 12 \cdot (3x - 2) \]

Calculating \(15 \cdot 8\):

\[ 120 = 12 \cdot (3x - 2) \]

Now, let's divide both sides of the equation by 12:

\[ 10 = 3x - 2 \]

Next, we add 2 to both sides:

\[ 12 = 3x \]

Now, we divide by 3:

\[ x = 4 \]

Thus, the solution is:

\[ \boxed{4} \]