Question

question 6 (1 point) the length of side hg is ______ blank 1

Explanation:

Step1: Identify Triangle Type

Triangle \( EFG \) has \( EF = FG = 14 \), so it's isosceles. \( FH \perp EG \), so \( \triangle EHF \) is right - angled with \( \angle E = 60^\circ \), making it a 30 - 60 - 90 triangle.

Step2: Find \( EH \) in \( \triangle EHF \)

In a 30 - 60 - 90 triangle, the side adjacent to \( 60^\circ \) ( \( EH \)) is half the hypotenuse when the hypotenuse is opposite \( 90^\circ \). Wait, no: in right - triangle \( \triangle EHF \), \( \cos(60^\circ)=\frac{EH}{EF} \). Since \( EF = 14 \) and \( \cos(60^\circ)=\frac{1}{2} \), then \( EH=EF\times\cos(60^\circ)=14\times\frac{1}{2} = 7 \).

Step3: Find \( HG \)

Since \( FH \) is the perpendicular bisector of \( EG \) (because \( \triangle EFG \) is isosceles with \( EF = FG \) and \( FH\perp EG \)), \( EH = HG \). So \( HG = 7 \).

Answer:

\( 7 \)