Question

match the special right triangle to its correct side ratios. 30° - 60° - 90° triangle choose 1:1:√2 1:√3:2 45° - 45° - 90° triangle 1:1:1 60° - 60° - 60° triangle choose

Explanation:

Step1: Recall properties of special triangles

For a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, if the side opposite \(30^{\circ}\) is \(x\), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse (opposite \(90^{\circ}\)) is \(2x\). So the ratio is \(1:\sqrt{3}:2\).
For a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, if the legs (sides opposite \(45^{\circ}\)) are \(x\) each, the hypotenuse (opposite \(90^{\circ}\)) is \(x\sqrt{2}\). So the ratio is \(1:1:\sqrt{2}\).
For a \(60^{\circ}-60^{\circ}-60^{\circ}\) triangle (equilateral triangle), all sides are equal. So the ratio is \(1:1:1\).

Answer:

• \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle: \(1:\sqrt{3}:2\)
• \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle: \(1:1:\sqrt{2}\)
• \(60^{\circ}-60^{\circ}-60^{\circ}\) triangle: \(1:1:1\)