Question

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

Explanation:

Step1: Analyze the 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 right - triangle, the hypotenuse is the longest side. So, \( EF \) is the hypotenuse, so \( EF \) is the longest side of \( \triangle DEF \).

Step2: Find the length of \( DF \)

In a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle, the side opposite the \( 30^{\circ} \) angle is half of the hypotenuse. The angle \( \angle E = 30^{\circ} \), and the side opposite to \( \angle E \) is \( DF \). So, \( DF=\frac{1}{2}EF \). Since \( EF = 12\space cm \), then \( DF=\frac{1}{2}\times12 = 6\space cm \).

Step3: Find the length of \( DE \)

In a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle, the side opposite the \( 60^{\circ} \) angle is \( \sqrt{3} \) times the side opposite the \( 30^{\circ} \) angle. The side opposite \( \angle F=60^{\circ} \) is \( DE \), and the side opposite \( \angle E = 30^{\circ} \) is \( DF = 6\space cm \). So, \( DE=\sqrt{3}\times DF \). Substituting \( DF = 6\space cm \), we get \( DE = 6\sqrt{3}\space cm \).

Answer:

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