Question

find the value of x. note that ab = 15, bc = 9, and cd = x.

Explanation:

Step1: Recall secant-tangent theorem

For a tangent segment $AB$ and a secant segment $BD$ intersecting the circle at $C$ and $D$, the theorem states: $AB^2 = BC \times BD$

Step2: Express BD in terms of x

Since $BD = BC + CD$, substitute $BC=9$, $CD=x$:
$BD = 9 + x$

Step3: Substitute values into theorem

Plug $AB=15$, $BC=9$, $BD=9+x$ into the formula:
$15^2 = 9 \times (9 + x)$

Step4: Simplify and solve for x

Calculate $15^2=225$, then:
$225 = 81 + 9x$
Subtract 81 from both sides:
$225 - 81 = 9x$
$144 = 9x$
Divide both sides by 9:
$x = \frac{144}{9} = 16$

Answer:

$x=16$