In $\triangle ABC$, $AD$ is the bisector of $\angle A$ meeting $BC$ at $D$. If $AB = 30$ cm, $BC = 20$ cm and the length of $BD$ is 4 cm less than $DC$, then the length (in cm) of $AC$ is

Given Information:

• Triangle $ABC$ with angle bisector $AD$ meeting $BC$ at $D$
• $AB = 30$ cm
• $BC = 20$ cm
• $BD$ is $4$ cm less than $DC$

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Since $BD$ is $4$ cm less than $DC$:

• Let $DC = x$ cm
• Then $BD = (x - 4)$ cm

Since $D$ lies on $BC$: $BD + DC = BC$

$(x - 4) + x = 20$

$2x - 4 = 20$

$2x = 24$

$x = 12$

Therefore: $DC = 12$ cm and $BD = 8$ cm

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When a line bisects an angle of a triangle, it divides the opposite side in the ratio of the adjacent sides.

The Angle Bisector Theorem states:

$\dfrac{BD}{DC} = \dfrac{AB}{AC}$

Substituting our values:

$\dfrac{8}{12} = \dfrac{30}{AC}$

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Cross multiply:

$8 \times AC = 12 \times 30$

$8 \times AC = 360$

$AC = \dfrac{360}{8}$

$AC = 45$ cm

Answer: $AC = 45$ cm