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\text{mm}\\) □ \\(\overline{ac} = 2ab\\) □ \\(\overline{ac} = 9\sqrt{3}\\) □ \\(\overline{bc}\\) is the longest segment in \\(\triangle abc\\)

Explanation:

Step1: Identify triangle type

Triangle \( ABC \) is a right - triangle with \( \angle B = 90^{\circ} \), \( \angle C=30^{\circ} \), \( \angle A = 60^{\circ} \), and \( AB = 9\space mm \). In a \( 30 - 60 - 90 \) triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \), where the side opposite \( 30^{\circ} \) ( \( AB \) in this case, since \( \angle C = 30^{\circ} \) and \( AB \) is opposite \( \angle C \)) is the shortest side, the side opposite \( 60^{\circ} \) ( \( BC \)) is \( \sqrt{3} \) times the shortest side, and the hypotenuse ( \( AC \)) is twice the shortest side.

Step2: Analyze each statement

• Statement 1: \( \overline{AB} \) is the shortest segment in \( \triangle ABC \). Since \( AB \) is opposite \( 30^{\circ} \), and in a \( 30 - 60 - 90 \) triangle, the side opposite \( 30^{\circ} \) is the shortest. So this statement is correct.
• Statement 2: \( \overline{BC}=18\space mm \). The length of \( BC \): In a \( 30 - 60 - 90 \) triangle, \( BC \) (opposite \( 60^{\circ} \)) \(=AB\times\sqrt{3}=9\sqrt{3}\space mm

eq18\space mm \). So this statement is incorrect.

• Statement 3: \( \overline{AC} = 2AB \). Since \( AC \) is the hypotenuse and in a \( 30 - 60 - 90 \) triangle, hypotenuse \( = 2\times \) side opposite \( 30^{\circ} \). Since \( AB \) is opposite \( 30^{\circ} \), \( AC=2AB = 2\times9 = 18\space mm \). So this statement is correct.
• Statement 4: \( \overline{AC}=9\sqrt{3} \). We know \( AC = 2AB=18\space mm

eq9\sqrt{3}\space mm \). So this statement is incorrect.

• Statement 5: \( \overline{BC} \) is the longest segment in \( \triangle ABC \). The hypotenuse \( AC \) is the longest side (since \( AC = 2AB \) and \( BC=9\sqrt{3}\approx15.59\space mm \), \( AC = 18\space mm \)). So this statement is incorrect. Also, let's re - check the length of \( BC \): \( BC=AB\times\sqrt{3}=9\sqrt{3}\space mm \), and \( AC = 18\space mm \), \( AB = 9\space mm \). So the longest side is \( AC \), not \( BC \). Wait, we made a mistake earlier. Wait, \( \angle C = 30^{\circ} \), so the side opposite \( \angle C \) is \( AB \), the side opposite \( \angle A=60^{\circ} \) is \( BC \), and the side opposite \( \angle B = 90^{\circ} \) is \( AC \). So:
• \( AB \) (opposite \( 30^{\circ} \)) \( = 9\space mm \)
• \( BC \) (opposite \( 60^{\circ} \)) \(=AB\times\sqrt{3}=9\sqrt{3}\space mm\approx15.59\space mm\)
• \( AC \) (opposite \( 90^{\circ} \)) \( = 2AB=18\space mm \)
• So the longest side is \( AC \), not \( BC \). So the statement " \( \overline{BC} \) is the longest segment in \( \triangle ABC \)" is incorrect. But let's re - check statement 4: \( AC = 18\space mm \), \( 9\sqrt{3}\approx15.59\space mm \), so statement 4 is incorrect. Wait, let's re - evaluate statement 3 and 1 again.
• Statement 1: \( AB \) is opposite \( 30^{\circ} \), so it is the shortest side. Correct.
• Statement 3: \( AC = 2AB \), since \( AC \) is hypotenuse and in \( 30 - 60 - 90 \) triangle, hypotenuse \( = 2\times \) shorter leg ( \( AB \) is shorter leg). So \( AC = 2\times9=18 = 2AB \). Correct.
• Also, let's check the length of \( BC \): \( BC=\sqrt{AC^{2}-AB^{2}}=\sqrt{18^{2}-9^{2}}=\sqrt{324 - 81}=\sqrt{243}=9\sqrt{3}\space mm \). So statement 4: \( AC = 9\sqrt{3} \) is incorrect (since \( AC = 18 \)), statement 2: \( BC = 18 \) is incorrect (since \( BC=9\sqrt{3}\approx15.59 \)). The statement " \( \overline{BC} \) is the longest segment" is incorrect because \( AC = 18 \) is longer than \( BC\approx15.59 \).

Answer:

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