图示两条弦 AB 和 CD 相交于 X，使得角 CXB = $90^\circ$。已知 $AC = 5\ \text{cm}$，$BD = 12\ \text{cm}$，且角 $BDC = \theta$。

1. Find angle BAC using circle properties.

Angles subtended by the same arc $BC$ at the circumference are equal.
$\angle BAC = \angle BDC = \theta$

2. Express the length of AX in terms of theta.

In $\triangle AXC$, $\angle AXC = 90^\circ$ (angles on a straight line with $\angle CXB = 90^\circ$).
$\cos(\angle CAX) = \frac{AX}{AC}$
$\cos\theta = \frac{AX}{5}$
$AX = 5\cos\theta$

3. Express the length of BX in terms of theta and sum them to find AB.

In $\triangle BXD$, $\angle BXD = 90^\circ$ (angles on a straight line with $\angle CXB = 90^\circ$).
$\sin(\angle BDX) = \frac{BX}{BD}$
$\sin\theta = \frac{BX}{12}$
$BX = 12\sin\theta$

Since $X$ lies on the chord $AB$,
$AB = AX + BX$
$AB = 5\cos\theta + 12\sin\theta$ (Shown)

1. Expand the target form and compare coefficients with the given expression

Let $5\cos\theta + 12\sin\theta = R\cos(\theta - \alpha)$.
Expanding the right side:
$R\cos(\theta - \alpha) = R\cos\theta\cos\alpha + R\sin\theta\sin\alpha$
Comparing coefficients of $\cos\theta$ and $\sin\theta$:
$R\cos\alpha = 5$
$R\sin\alpha = 12$

2. Calculate the value of R

$R = \sqrt{5^2 + 12^2}$
$R = \sqrt{25 + 144} = \sqrt{169}$
$R = 13$

3. Calculate the value of alpha

$\tan\alpha = \frac{R\sin\alpha}{R\cos\alpha} = \frac{12}{5}$
$\alpha = \tan^{-1}\left(\frac{12}{5}\right)$
$\alpha \approx 67.380^\circ \approx 67.4^\circ$ (to 1 d.p.)

4. State the final expression

$AB = 13\cos(\theta - 67.4^\circ)$

1. State the maximum length of AB using the R-formula expression

From part (b), $AB = 13\cos(\theta - 67.38^\circ)$.
The maximum value of $\cos(\theta - 67.38^\circ)$ is $1$.
Maximum length of $AB = 13(1) = 13\text{ cm}$.

2. Find the corresponding value of $\theta$

For maximum $AB$, $\cos(\theta - 67.38^\circ) = 1$.
$\theta - 67.38^\circ = 0^\circ$
$\theta = 67.38^\circ \approx 67.4^\circ$ (to 1 d.p.)

3. State the radius of the circle

The maximum length of a chord in a circle is its diameter.
Diameter $= 13\text{ cm}$
Radius $= \frac{13}{2} = 6.5\text{ cm}$