Question

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

Explanation:

Step1: Identify triangle type

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

Step2: Find the length of \( DF \) (opposite \( 30^{\circ}\))

The side opposite \( \angle E = 30^{\circ}\) is \( DF\). Let the length of \( DF=x\). Since the hypotenuse \( EF = 2x\) and \( EF = 12\space cm\), we have \( 2x=12\), so \( x = 6\space cm\). So \( DF = 6\space cm\) (this statement is correct).

Step3: Find the length of \( DE \) (opposite \( 60^{\circ}\))

The side opposite \( \angle F=60^{\circ}\) is \( DE\). Using the ratio, if \( DF=x = 6\space cm\), then \( DE=x\sqrt{3}=6\sqrt{3}\space cm\) (this statement is correct).

Step4: Check the longest side

In a right - triangle, the hypotenuse is the longest side. Here, the hypotenuse is \( EF\), so \( EF\) is the longest side of \( \triangle DEF\) (this statement is correct).

Step5: Analyze incorrect statements

• For \( DE = 12\sqrt{3}\space cm\): From step 3, \( DE = 6\sqrt{3}\space cm\), so this is incorrect.
• For \( DF = 4\sqrt{3}\space cm\): From step 2, \( DF = 6\space cm\), so this is incorrect.

Answer:

The correct statements are:

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