Question

select all the correct answers. which three pairs of side - lengths are possible measurements for the triangle? gh = 2√3, hi = 2 gh = 8, gi = 16 hi = 7, gi = 14 gh = 8√2, gi = 16 gh = 2, hi = 2√3 gh = 8√3, gi = 16

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratios

In a 30 - 60 - 90 right - triangle, if the shorter leg (opposite the 30° angle) is $a$, the longer leg (opposite the 60° angle) is $a\sqrt{3}$, and the hypotenuse is $2a$. Let $HI$ be the shorter leg (opposite the 30° angle), $GH$ be the longer leg (opposite the 60° angle), and $GI$ be the hypotenuse.

Step2: Check each option

• Option 1: If $HI = 2$, then $GH=2\sqrt{3}$ and $GI = 4$. This option is incorrect.
• Option 2: If $GH = 8$, then $HI=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3}$ and $GI=\frac{16}{\sqrt{3}}=\frac{16\sqrt{3}}{3}$. This option is incorrect.
• Option 3: If $HI = 7$, then $GH = 7\sqrt{3}$ and $GI=14$. This option is correct.
• Option 4: If $GH = 8\sqrt{2}$, then $HI=\frac{8\sqrt{2}}{\sqrt{3}}=\frac{8\sqrt{6}}{3}$ and $GI=\frac{16\sqrt{2}}{\sqrt{3}}=\frac{16\sqrt{6}}{3}$. This option is incorrect.
• Option 5: If $GH = 2$, then $HI=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$ and $GI=\frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3}$. This option is incorrect.
• Option 6: If $GH = 8\sqrt{3}$, then $HI = 8$ and $GI = 16$. This option is correct.

Answer:

HI = 7, GI = 14
GH = 8\sqrt{3}, GI = 16