Question

the length of segment ab is 9 mm.
which statements regarding triangle abc are correct?
check all that apply.
$overline{ab}$ is the shortest segment in $\triangle abc$.
$overline{bc}$ = 18mm
$overline{ac}$ = 2$overline{ab}$
$overline{ac}$ = $9\sqrt{3}$
$overline{bc}$ is the longest segment in $\triangle abc$

Explanation:

Step1: Identify triangle type

$\triangle ABC$ is a 30-60-90 right triangle, right-angled at $B$, with $AB=9$ mm, $\angle C=30^\circ$, $\angle A=60^\circ$.

Step2: Relate sides to angles

In a triangle, the shortest side is opposite the smallest angle, longest side opposite largest angle. $\angle C=30^\circ$ (smallest) opposite $AB$; $\angle A=60^\circ$ opposite $BC$; $\angle B=90^\circ$ (largest) opposite $AC$.

Step3: Calculate $BC$ length

Use $\tan(30^\circ)=\frac{AB}{BC}$. Rearrange: $BC=\frac{AB}{\tan(30^\circ)}=\frac{9}{\frac{1}{\sqrt{3}}}=9\sqrt{3}$ mm.

Step4: Calculate $AC$ length

Use $\cos(30^\circ)=\frac{BC}{AC}$ or $AC=2AB$ (30-60-90 triangle property: hypotenuse = 2×short leg). $AC=2\times9=18$ mm.

Step5: Verify each statement

1. $\overline{AB}$ is shortest: True (opposite $30^\circ$).
2. $\overline{BC}=18$ mm: False ($BC=9\sqrt{3}$ mm).
3. $\overline{AC}=2\overline{AB}$: True ($18=2\times9$).
4. $\overline{AC}=9\sqrt{3}$: False ($AC=18$ mm).
5. $\overline{BC}$ is longest: False (longest is $AC$).

Answer:

• $\overline{AB}$ is the shortest segment in $\triangle ABC$.
• $\overline{AC} = 2\overline{AB}$