Question

what is bd? bd =

Explanation:

Step1: Apply the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if \(AC\) is the angle bisector of \(\angle BAD\), then \(\frac{AB}{AD}=\frac{BC}{CD}\). Given \(AB = 12\), \(AD=27\), \(BC = x\), and \(CD=x + 5\). Substitute these values into the theorem: \(\frac{12}{27}=\frac{x}{x + 5}\).

Step2: Cross - multiply the proportion

Cross - multiplying gives \(12(x + 5)=27x\). Expand the left - hand side: \(12x+60 = 27x\).

Step3: Solve for \(x\)

Subtract \(12x\) from both sides: \(60=27x - 12x\). So, \(15x = 60\), and \(x=\frac{60}{15}=4\).

Step4: Calculate \(BD\)

Since \(BD=BC + CD=x+(x + 5)\), substitute \(x = 4\). Then \(BD=4+(4 + 5)=13\).

Answer:

\(13\)