Question

which triangle is a $30^{circ}$-$60^{circ}$-$90^{circ}$ triangle?

• first triangle: sides 10, 5, $5sqrt{3}$
• second triangle: sides 15, 5, $5sqrt{3}$
• third triangle: sides 10, 5, $10sqrt{3}$
• fourth triangle: sides 15, 10, $5sqrt{3}$

Explanation:

Step1: Recall 30-60-90 side rules

In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, if the shortest leg (opposite $30^\circ$) is $s$, the longer leg (opposite $60^\circ$) is $s\sqrt{3}$, and the hypotenuse (opposite $90^\circ$) is $2s$.

Step2: Check first triangle sides

Sides: $5$, $5\sqrt{3}$, $10$. Here $s=5$, so $2s=10$, $s\sqrt{3}=5\sqrt{3}$. This matches the rule.

Step3: Verify it is right triangle

Use Pythagorean theorem: $5^2 + (5\sqrt{3})^2 = 25 + 75 = 100 = 10^2$. It satisfies the theorem.

Step4: Eliminate other options

• Second triangle: $5^2 + (5\sqrt{3})^2 = 100

eq 15^2=225$

• Third triangle: $5^2 + (10\sqrt{3})^2 = 25 + 300 = 325

eq 10^2=100$

• Fourth triangle: $10^2 + (5\sqrt{3})^2 = 100 + 75 = 175

eq 15^2=225$

Answer:

The triangle with side lengths 5, $5\sqrt{3}$, and 10 (the first option)