Question

angles: practice
select all the correct answers.
which given pairs of side lengths are possible measurements for the triangle?
$ab = 3, bc = 3\sqrt{3}$
$bc = 19\sqrt{3}, ac = 38\sqrt{3}$
$ab = 4\sqrt{3}, bc = 4$
$ab = 14, ac = 28$
$bc = 5\sqrt{2}, ac = 10$
$bc = 18, ac = 12$

Explanation:

Step1: Identify triangle ratios

In a 30-60-90 right triangle, the side ratios are:
$AB : BC : AC = 1 : \sqrt{3} : 2$
where $AB$ = side opposite 30°, $BC$ = side opposite 60°, $AC$ = hypotenuse.

Step2: Verify Option 1

Check $AB=3, BC=3\sqrt{3}$:
$\frac{AB}{BC} = \frac{3}{3\sqrt{3}} = \frac{1}{\sqrt{3}}$, matches the ratio.

Step3: Verify Option 2

Check $BC=19\sqrt{3}, AC=38\sqrt{3}$:
$\frac{BC}{AC} = \frac{19\sqrt{3}}{38\sqrt{3}} = \frac{1}{2}$, but correct ratio for $BC:AC$ is $\frac{\sqrt{3}}{2}$. Does not match.

Step4: Verify Option 3

Check $AB=4\sqrt{3}, BC=4$:
$\frac{AB}{BC} = \frac{4\sqrt{3}}{4} = \sqrt{3}$, but correct ratio is $\frac{1}{\sqrt{3}}$. Does not match.

Step5: Verify Option 4

Check $AB=14, AC=28$:
$\frac{AB}{AC} = \frac{14}{28} = \frac{1}{2}$, matches the ratio of short side to hypotenuse.

Step6: Verify Option 5

Check $BC=5\sqrt{3}, AC=10$:
$\frac{BC}{AC} = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}$, matches the ratio of longer leg to hypotenuse.

Step7: Verify Option 6

Check $BC=18, AC=12$:
Hypotenuse $AC$ cannot be shorter than leg $BC$. Does not match.

Answer:

• AB = 3, BC = $3\sqrt{3}$
• AB = 14, AC = 28
• BC = $5\sqrt{3}$, AC = 10