Given m( C) = 54^ 13'. To convert minutes to degrees, divide by 60.

Given $m(\angle C) = 54^\circ 13'$. To convert minutes to degrees, divide by 60.

$$13' = \frac{13}{60}^\circ \approx 0.2167^\circ$$

So, $m(\angle C) \approx 54.2167^\circ$.

In a right-angled triangle $\triangle ABC$ with the right angle at B:

• The side opposite to $\angle C$ is AB.
• The side adjacent to $\angle C$ is BC. The tangent function relates the opposite and adjacent sides:

$$\tan(\angle C) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC}$$

We are given $m(\angle C) = 54^\circ 13'$ and $BC = 20$ cm.

$$\tan(54^\circ 13') = \frac{AB}{20}$$

Multiply both sides by 20:

$$AB = 20 \times \tan(54^\circ 13')$$ $$AB = 20 \times \tan(54.2167^\circ)$$

Using a calculator:

$$AB \approx 20 \times 1.3868$$ $$AB \approx 27.736$$

Rounding to one decimal place, the length of AB is approximately 27.7 cm.

The length of AB is $\text{27.7 cm}$.