Question

which triangle is a 30° - 60° - 90° triangle? 10 5 5√3 15 5 5√3 10 5 10√3 15 10 5√3

Explanation:

Step1: Recall the side - ratio of \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the side - ratio is \(1:\sqrt{3}:2\) (shorter leg : longer leg : hypotenuse).

Step2: Check each option

• First option:

Let the shorter leg \(a = 5\). Then the hypotenuse \(c=2a\). If \(a = 5\), then \(c = 10\). The longer leg \(b=a\sqrt{3}\), if \(a = 5\), then \(b = 5\sqrt{3}\).

• Second option:

If the shorter leg \(a = 5\), the hypotenuse should be \(c = 2a=10\), but here \(c = 15\).

• Third option:

If the shorter leg \(a = 5\), the longer leg should be \(b=a\sqrt{3}=5\sqrt{3}\), but here \(b = 10\sqrt{3}\).

• Fourth option:

If the shorter leg \(a\) (assuming \(a = 5\sqrt{3}\) is not correct as per the ratio. If we assume \(a = 10\), the hypotenuse should be \(c = 20\) (not \(15\))

So, the triangle with sides \(5\), \(10\), \(5\sqrt{3}\) is a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle.

Answer:

First option (triangle with sides \(5\), \(10\), \(5\sqrt{3}\))