Question

the length of segment ef is 12 cm.
which statements regarding triangle def are correct? choose three correct answers.
df = 4√3 cm
de = 12√3 cm
de = 6√3 cm
ef is the longest side of △def.
df = 6 cm

Explanation:

Step1: Analyze triangle DEF

Triangle \( DEF \) is a right - triangle with \( \angle D = 90^{\circ} \), \( \angle E=30^{\circ} \), \( \angle F = 60^{\circ} \) and \( EF = 12\space cm \). In a 30 - 60 - 90 triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \), where the side opposite \( 30^{\circ} \) is the shortest side (let's call it \( x \)), the side opposite \( 60^{\circ} \) is \( x\sqrt{3} \) and the hypotenuse (opposite \( 90^{\circ} \)) is \( 2x \).

Step2: Find the length of DF

The side \( DF \) is opposite \( \angle E = 30^{\circ} \). Let \( DF=x \). The hypotenuse \( EF = 2x \). Given \( EF = 12\space cm \), so \( 2x=12 \), then \( x = DF=6\space cm \).

Step3: Find the length of DE

The side \( DE \) is opposite \( \angle F=60^{\circ} \). Since \( DF = 6\space cm \), by the ratio of 30 - 60 - 90 triangle, \( DE=DF\times\sqrt{3}=6\sqrt{3}\space cm \).

Step4: Analyze the longest side

In a right - triangle, the hypotenuse is the longest side. The hypotenuse of \( \triangle DEF \) is \( EF \) (because \( \angle D = 90^{\circ} \)), so \( EF \) is the longest side of \( \triangle DEF \).

Now let's check each statement:

• Statement 1: \( DF = 6\space cm \) (correct, as we found \( DF = 6\space cm \))
• Statement 2: \( EF \) is the longest side of \( \triangle DEF \) (correct, because \( EF \) is the hypotenuse)
• Statement 3: \( DE = 6\sqrt{3}\space cm \) (correct, from step 3)
• Statement 4: \( DE=12\sqrt{3}\space cm \) (incorrect, since we found \( DE = 6\sqrt{3}\space cm \))
• Statement 5: \( DF = 4\sqrt{3}\space cm \) (incorrect, since we found \( DF = 6\space cm \))

Answer:

The correct statements are:

• \( DF = 6\space cm \)
• \( EF \) is the longest side of \( \triangle DEF \)
• \( DE = 6\sqrt{3}\space cm \)