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: Identify triangle type

$\triangle DEF$ is a right triangle with $\angle D=90^\circ$, hypotenuse $EF=12$ cm, $\angle E=30^\circ$, $\angle F=60^\circ$.

Step2: Check longest side

In right triangles, hypotenuse is longest.
$EF$ is hypotenuse, so it is the longest side.

Step3: Calculate DF (30° opposite side)

In 30-60-90 triangle, side opposite $30^\circ$ is $\frac{1}{2}$ hypotenuse.
$\angle E=30^\circ$, opposite side $DF=\frac{1}{2} \times 12 = 6$ cm

Step4: Calculate DE (60° opposite side)

Side opposite $60^\circ$ is $\frac{\sqrt{3}}{2}$ hypotenuse.
$DE=\frac{\sqrt{3}}{2} \times 12 = 6\sqrt{3}$ cm

Answer:

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