Identify special right triangles - High school Trigonometry Exercises

Determine whether the following right triangles are special or not.

This is not a special right triangle.

This is a 30°−60°−90° triangle.

This is a 45°−45°−90° triangle.

Three facts are given:

One angle measurement
Two side lengths, with one side length being the hypotenuse (that is opposite of the right angle).

By knowing the ratios of side lengths for special triangles, and the side lengths of the hypotenuse and one other side length, the ratio of side lengths in this triangle can be compared with the known ratios for special triangles.

In this triangle, the hypotenuse's length is 2 and one of the other sides' length is \sqrt{2}. Their ratio is:

\dfrac{2}{\sqrt{2}}

By recognizing that 2 = \sqrt{2}^2, the ratio can be re-written as:

\dfrac{2}{\sqrt{2}}=\dfrac{\sqrt{2}^2}{\sqrt{2}} = \dfrac{\sqrt{2}}{1}=\sqrt{2}

This is the ratio of hypotenuse to non-hypotenous side lengths in a right isosceles triangle.

This is a 45°-45°-90° triangle.

This is a 45°−45°−90° triangle.

This is not a special right triangle.

This is a 30°−60°−90° triangle.

Three facts are given:

One angle measurement
Two side lengths, with one side length being the hypotenuse (that is opposite of the right angle).

By knowing the ratios of side lengths for special triangles, and the side lengths of the hypotenuse and one other side, the ratio of side lengths in this triangle can be compared with the known ratios for special triangles.

In this triangle, the hypotenuse's length is 3 and one of the other sides' length is \sqrt{3}. Their ratio is:

\dfrac{3}{\sqrt{3}}

By recognizing that 3 = \sqrt{3}^2, the ratio can be re-written as:

\dfrac{3}{\sqrt{3}} = \dfrac{\sqrt{3}^2}{\sqrt{3}} = \dfrac{\sqrt{3}}{1} = \sqrt{3}

This is not the ratio of hypotenuse to non-hypotenuse side lengths in a right isosceles triangle (\sqrt{2}:1), so this is not a 45^{\circ}-45^{\circ}-90^{\circ} triangle.

This is also not the ratio of the hypotenuse to either the shorter or longer non-hypotenuse side of a 30^{\circ}-60^{\circ}-90^{\circ} triangle ( 2:1 and 2:\sqrt{3}, respectively), so this is not a 30^{\circ}-60^{\circ}-90^{\circ} triangle.

This is not a special triangle.

This is a 45°−45°−90° triangle.

This is a 30°−60°−90° triangle.

This is not a special right triangle.

Three facts are given:

One angle measurement
Two side lengths, with one side length being the hypotenuse (that is opposite of the right angle).

In this triangle, the hypotenuse's length is 4 and one of the other sides' length is 2\sqrt{2}. Their ratio is:

\dfrac{4}{2\sqrt{2}}

By recognizing that 4 = 2\sqrt{2}^2, the ratio can be re-written as:

\dfrac{2\sqrt{2}^2}{2\sqrt{2}} = \sqrt{2}

This is the ratio of hypotenuse to non-hypotenuse side lengths in a right isosceles triangle.

This is a 45°-45°-90° triangle.

This is a 30°−60°−90° triangle.

This is a 45°−45°−90° triangle.

This is not a special right triangle.

Three facts are given:

One angle measurement
Two side lengths, with one side length being the hypotenuse (that is opposite of the right angle).

In this triangle, the hypotenuse's length is 2\sqrt{3} and one of the other sides' length is 3. Their ratio is:

\dfrac{2\sqrt{3}}{3}

By recognizing that 3 = \sqrt{3}^2, the ratio can be re-written as:

\dfrac{2\sqrt{3}}{3} = \dfrac{2\sqrt{3}}{\sqrt{3}^2} = \dfrac{2}{\sqrt{3}}

This is the ratio of the hypotenuse to the longer non-hypotenuse side of a 30^{\circ}-60^{\circ}-90^{\circ} triangle.

This is a 30^{\circ}-60^{\circ}-90^{\circ} triangle

This is a 30°−60°−90° triangle.

This is a 45°−45°−90° triangle.

This is not a special right triangle.

Three facts are given:

One angle measurement
The lengths of both the non-hypotenuse sides (the sides adjacent to the right angle).

In this triangle, the longer and shorter non-hypotenuse side lengths are 3 and \sqrt{2}, respectively. Their ratio is:

\dfrac{3}{\sqrt{2}}

This is not the ratio of the non-hypotenuse side lengths in a right isosceles triangle (1:1), so this is not a 45^{\circ}-45^{\circ}-90^{\circ} triangle.

This is also not the ratio of the length of the longer non-hypotenuse side to the length of the shorter non-hypotenuse side in a 30^{\circ}-60^{\circ}-90^{\circ} triangle (\sqrt{3}:1), so this is not a 30^{\circ}-60^{\circ}-90^{\circ} triangle.

This is not a special triangle.

This is not a special right triangle.

This is a 45°−45°−90° triangle.

This is a 30°−60°−90° triangle.

Three facts are given:

One angle measurement
Two side lengths, with one side length being the hypotenuse (that is opposite of the right angle).

In this triangle, the hypotenuse's side length is 4 and one of the other sides' length is \sqrt{2}. Their ratio is:

\dfrac{4}{\sqrt{2}}

By recognizing that 4 = 2\sqrt{2}^2, the ratio can be re-written as:

\dfrac{4}{\sqrt{2}} = \dfrac{2\sqrt{2}^2}{\sqrt{2}} = \dfrac{2\sqrt{2}}{1} = 2\sqrt{2}

This is not the ratio of the hypotenuse to non-hypotenuse side lengths in a right isosceles triangle (\sqrt{2}:1), so this is not a 45^{\circ}-45^{\circ}-90^{\circ} triangle.

This is also not the ratio of the hypotenuse to non-hypotenuse side lengths in a 30^{\circ}-60^{\circ}-90^{\circ} triangle (\sqrt{3}:1), so this is not a 30^{\circ}-60^{\circ}-90^{\circ} triangle.

This is not a special triangle.

This is not a special right triangle.

This is a 30°−60°−90° triangle.

This is a 45°−45°−90° triangle.

Three facts are given:

One angle measurement
The lengths of both the non-hypotenuse sides (the sides adjacent to the right angle).

By knowing the ratios of side lengths for special triangles, and the side lengths of both non-hypotenuse sides, the ratio of side lengths in this triangle can be compared with the known ratios for special triangles.

In this triangle, the longer and shorter non-hypotenuse side lengths are 2\sqrt{3} and 2, respectively. Their ratio is:

\dfrac{2\sqrt{3}}{2}

This is the ratio of the lengths of the longer non-hypotenuse side to the shorter non-hypotenuse side of a 30^{\circ}-60^{\circ}-90^{\circ} triangle.

This is a 30^{\circ}-60^{\circ}-90^{\circ} triangle.