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 = 12\sqrt{3}\\) cm
□ \\(df = 4\sqrt{3}\\) cm
□ \\(de = 6\sqrt{3}\\) 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 hypotenuse \( EF = 12\space\text{cm} \). In a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \), where the side opposite \( 30^{\circ} \) (the shorter leg) is \( \frac{1}{2} \) of the hypotenuse, the side opposite \( 60^{\circ} \) (the longer leg) is \( \frac{\sqrt{3}}{2} \) of the hypotenuse, and the hypotenuse is the longest side.

Step2: Check " \( \overline{EF} \) is the longest side of \( \triangle DEF \)"

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: Check " \( DF = 6\space\text{cm} \)"

The side \( DF \) is opposite \( \angle E = 30^{\circ} \). In a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle, the side opposite \( 30^{\circ} \) is \( \frac{1}{2}\times \text{hypotenuse} \). So \( DF=\frac{1}{2}\times EF \). Substituting \( EF = 12\space\text{cm} \), we get \( DF=\frac{1}{2}\times12 = 6\space\text{cm} \). This statement is correct.

Step4: Check " \( DE = 12\sqrt{3}\space\text{cm} \)"

The side \( DE \) is opposite \( \angle F=60^{\circ} \). In a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle, the side opposite \( 60^{\circ} \) is \( \frac{\sqrt{3}}{2}\times\text{hypotenuse} \). So \( DE=\frac{\sqrt{3}}{2}\times EF \). Substituting \( EF = 12\space\text{cm} \), we get \( DE=\frac{\sqrt{3}}{2}\times12=6\sqrt{3}\space\text{cm}
eq12\sqrt{3}\space\text{cm} \). So this statement is incorrect.

Step5: Check " \( DF = 4\sqrt{3}\space\text{cm} \)"

We already found that \( DF = 6\space\text{cm} \) (from Step 3), and \( 6
eq4\sqrt{3}\) (since \( 4\sqrt{3}\approx4\times1.732 = 6.928
eq6 \)). So this statement is incorrect.

Step6: Check " \( DE = 6\sqrt{3}\space\text{cm} \)"

As calculated in Step 4, \( DE=\frac{\sqrt{3}}{2}\times12 = 6\sqrt{3}\space\text{cm} \). This statement is correct.

Answer:

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