Question

segment $overline{bd}$ bisects $angle abc$. solve for $x$. round to the nearest tenth, if necessary. (image not necessarily to scale.)

answer attempt 2 out of 2
$x = \boxed{9}$
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Explanation:

Step1: Apply Angle Bisector Theorem

The Angle Bisector Theorem states that if a bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. So, $\frac{AD}{DC} = \frac{AB}{BC}$.
Here, $AD = x$, $DC = 9$, $AB = 16$, $BC = 12$.
So, $\frac{x}{9} = \frac{16}{12}$.

Step2: Solve for \( x \)

Cross - multiply: $12x = 16\times9$.
Calculate $16\times9 = 144$.
Then $x=\frac{144}{12}=12$. Wait, no, wait. Wait, the sides: $BC = 12$, $AB = 16$, $DC = 9$, $AD = x$. Wait, maybe I mixed up the sides. Wait, the Angle Bisector Theorem is $\frac{AD}{DC}=\frac{AB}{BC}$. Wait, $AB = 16$, $BC = 12$, $DC = 9$, $AD = x$. So $\frac{x}{9}=\frac{16}{12}$. Simplify $\frac{16}{12}=\frac{4}{3}$. Then $x = 9\times\frac{4}{3}=12$? Wait, but maybe the sides are $AC$ is split into $DC = 9$ and $AD = x$, so $AC=9 + x$. And the other sides: $AB = 16$, $BC = 12$. Wait, no, the Angle Bisector Theorem is $\frac{AD}{DC}=\frac{AB}{BC}$. So $AD=x$, $DC = 9$, $AB = 16$, $BC = 12$. So $\frac{x}{9}=\frac{16}{12}$. Cross - multiply: $12x=16\times9 = 144$. Then $x = 144\div12 = 12$. Wait, but maybe I had the sides wrong. Wait, let's check again. The triangle is $ABC$, with $B$ connected to $D$ on $AC$. So $BD$ bisects $\angle ABC$, so by Angle Bisector Theorem, $\frac{AB}{BC}=\frac{AD}{DC}$. So $AB = 16$, $BC = 12$, $AD = x$, $DC = 9$. So $\frac{16}{12}=\frac{x}{9}$. Simplify $\frac{16}{12}=\frac{4}{3}$. So $\frac{4}{3}=\frac{x}{9}$. Cross - multiply: $3x=4\times9 = 36$. Then $x = 12$. Wait, that's correct. So the previous attempt was wrong. Let's do it again.

Step1: Recall Angle Bisector Theorem

The Angle Bisector Theorem states that if a bisector of an angle of a triangle divides the opposite side into two segments, then the ratio of the lengths of these two segments is equal to the ratio of the lengths of the other two sides of the triangle. So, for $\triangle ABC$ with angle bisector $BD$ (bisecting $\angle ABC$ and intersecting $AC$ at $D$), we have $\frac{AD}{DC}=\frac{AB}{BC}$.

Step2: Identify the lengths

We know that $AB = 16$, $BC = 12$, $DC = 9$, and $AD = x$.

Step3: Set up the proportion

Substitute the known values into the proportion from the Angle Bisector Theorem: $\frac{x}{9}=\frac{16}{12}$.

Step4: Simplify the right - hand side

Simplify $\frac{16}{12}=\frac{4}{3}$. So the equation becomes $\frac{x}{9}=\frac{4}{3}$.

Step5: Solve for \( x \)

Cross - multiply: $3x=4\times9$.
Calculate $4\times9 = 36$.
Then $x=\frac{36}{3}=12$.

Answer:

$x = 12$