Question

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

Explanation:

Step1: Analyze 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: Check "EF is the longest side"

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

Step3: Calculate \( DF \)

The side \( DF \) is opposite \( \angle E = 30^{\circ} \). Let the length of \( DF=x \). We know that the hypotenuse \( EF = 2x \) (from the \( 30 - 60 - 90 \) triangle ratio). Given \( EF = 12\space cm \), so \( 2x=12\), then \( x = 6\space cm \). So \( DF = 6\space cm \), this statement is correct.

Step4: Calculate \( DE \)

The side \( DE \) is opposite \( \angle F=60^{\circ} \). From the \( 30 - 60 - 90 \) triangle ratio, the side opposite \( 60^{\circ} \) is \( x\sqrt{3} \), where \( x = DF = 6\space cm \). So \( DE=6\sqrt{3}\space cm \). Let's verify using trigonometry: \( \cos(30^{\circ})=\frac{DE}{EF} \), \( DE = EF\times\cos(30^{\circ})=12\times\frac{\sqrt{3}}{2}=6\sqrt{3}\space cm \). So the statement \( DE = 6\sqrt{3}\space cm \) is correct, and \( DE = 12\sqrt{3}\space cm \) is incorrect. Also, \( DF = 4\sqrt{3}\space cm \) is incorrect.

Answer:

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