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} = 18)mm
□ (overline{ac} = 2overline{ab})
□ (overline{ac} = 9sqrt{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 $B$. In such triangles, sides follow: shortest side (opposite 30°) = $\frac{1}{2}$ hypotenuse, longer leg (opposite 60°) = $\text{shortest side} \times \sqrt{3}$.

Step2: Match sides to angles

• $\angle C = 30^\circ$, opposite side $\overline{AB} = 9$ mm (shortest side)
• $\angle A = 60^\circ$, opposite side $\overline{BC}$
• $\angle B = 90^\circ$, opposite side $\overline{AC}$ (hypotenuse, longest side)

Step3: Calculate $\overline{AC}$

Hypotenuse = $2 \times$ shortest side:
$\overline{AC} = 2 \times \overline{AB} = 2 \times 9 = 18$ mm

Step4: Calculate $\overline{BC}$

Longer leg = $\text{shortest side} \times \sqrt{3}$:
$\overline{BC} = 9 \times \sqrt{3} = 9\sqrt{3}$ mm

Step5: Verify each statement

1. $\overline{AB}$ is shortest: True (9 mm < $9\sqrt{3}$ < 18)
2. $\overline{BC}=18$ mm: False ($\overline{BC}=9\sqrt{3}$)
3. $\overline{AC}=2\overline{AB}$: True (18 = 2×9)
4. $\overline{AC}=9\sqrt{3}$: False ($\overline{AC}=18$)
5. $\overline{BC}$ is longest: False ($\overline{AC}$ is longest)

Answer:

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