Question

In $\triangle ABC$, $AD$ is the internal bisector of $\angle A$, meeting the side $BC$ at $D$. If $BD = 5 \ \mathrm{cm}$ and $BC = 7.5 \ \mathrm{cm}$, then $AB:AC$ is: A. $1:2$ B. $2:1$ C. $3:1$ D. $1:3$

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Answer

To solve the problem, let's consider the properties of an angle bisector in a triangle. Here, $AD$ bisects $\angle A$ in $\triangle ABC$. According to the Angle Bisector Theorem, the angle bisector divides the opposite side into two segments that are proportional to the adjacent sides.

From the question, we know: $$ BD = 5 \ \text{cm} $$ $$ BC = 7.5 \ \text{cm} $$ Since $BC = BD + DC$, we can find $DC$ as: $$ DC = BC - BD = 7.5 \ \text{cm} - 5 \ \text{cm} = 2.5 \ \text{cm} $$

According to the Angle Bisector Theorem: $$ \frac{AB}{AC} = \frac{BD}{DC} $$ Substituting the values we have: $$ \frac{AB}{AC} = \frac{5 \ \text{cm}}{2.5 \ \text{cm}} = 2 $$

This means $AB$ is twice as long as $AC$. Therefore, the ratio of $AB:AC$ is: $$ 2:1 $$

The correct answer is Option B: $2:1$.


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