Question

$\overline{ax}$ and $\overline{ex}$ are secant segments that intersect at point x.
what is the length of $\overline{de}$?
1 unit
3 units
$4\frac{1}{2}$ units
$4\frac{2}{3}$ units

Explanation:

Step1: Recall secant segment theorem

If two secant segments are drawn from a point outside a circle, then $XB \times XA = XD \times XE$.

Step2: Define unknown length

Let $DE = x$. Then $XE = XD + DE = 3 + x$. Calculate $XA = AB + BX = 7 + 2 = 9$, $XB = 2$, $XD = 3$.

Step3: Substitute into theorem formula

$$2 \times 9 = 3 \times (3 + x)$$

Step4: Simplify and solve for x

Calculate left side: $18 = 9 + 3x$
Subtract 9 from both sides: $9 = 3x$
Divide by 3: $x = 3$

Answer:

3 units