Question

which triangle is a 30°-60°-90° triangle?
10
5
$5\sqrt{3}$
15
5
$5\sqrt{3}$
10
5
$10\sqrt{3}$
15
10
$5\sqrt{3}$

Explanation:

Step1: Recall 30-60-90 triangle ratios

In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the side lengths are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^\circ\) is the shortest (\(x\)), opposite \(60^\circ\) is \(x\sqrt{3}\), and hypotenuse is \(2x\).

Step2: Check each option

• First triangle: Sides \(5\), \(5\sqrt{3}\), \(10\). Let \(x = 5\). Then \(x\sqrt{3}=5\sqrt{3}\), \(2x = 10\). Matches the ratio \(1:\sqrt{3}:2\) (since \(5:5\sqrt{3}:10 = 1:\sqrt{3}:2\) when divided by 5).
• Second triangle: Sides \(5\), \(5\sqrt{3}\), \(15\). If \(x = 5\), \(2x = 10

eq15\). Doesn't match.

• Third triangle: Sides \(5\), \(10\sqrt{3}\), \(10\). If \(x = 5\), \(x\sqrt{3}=5\sqrt{3}

eq10\sqrt{3}\). Doesn't match.

• Fourth triangle: Sides \(10\), \(5\sqrt{3}\), \(15\). If \(x = 5\), \(2x = 10\), but hypotenuse should be \(2x\), here hypotenuse would be 15, but \(2x = 10

eq15\). Doesn't match.

Answer:

The first triangle (with sides 5, \(5\sqrt{3}\), 10)