# Special Right Triangles

Trigonometry: Special Right Triangles

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• There are 2 types of special right triangles.
Type 1. Triangle that is half of a equilateral triangle. Its 3 angle measures are: 30, 60 and 90 deg. Its side measures are : a, a/2; and (a*sqr.3)/2.
Type 2. Triangle that has its side measures in the ratio of 3:4:5. The proof is given by the Pythagor theorem: c^2 = b^2 + a^2.
Use of special right triangles.
In the old time, people use the special right triangles with sides ratio 3:4:5 to figure out, in the field, a right angle or a rectangular, or square, shape.
Now, students just use the properties of special right triangle to find, by computing, the unknown sides or angles.

Consider the properties of the sides, the angles and the symmetry.

#### Explanation:

$45 - 45 - 90 \text{ }$ refers to the angles of the triangle.

The color(blue)("sum of the angles is " 180°)

There are $\textcolor{b l u e}{\text{two equal angles}}$, so this is an isosceles triangle.

It therefore also has $\textcolor{b l u e}{\text{ two equal sides.}}$

The third angle is 90°. It is a $\textcolor{b l u e}{\text{right-angled triangle}}$ therefore Pythagoras' Theorem can be used.

The $\textcolor{b l u e}{\text{sides are in the ratio } 1 : 1 : \sqrt{2}}$

It has $\textcolor{b l u e}{\text{one line of symmetry}}$ - the perpendicular bisector of the base (the hypotenuse) passes through the vertex, (the 90° angle).

It has $\textcolor{b l u e}{\text{no rotational symmetry.}}$

• $m a t h b f \left\{{30}^{\circ} \text{-"60^circ"-} {90}^{\circ}\right\}$ Triangle

The ratios of three sides of a ${30}^{\circ} \text{-"60^circ"-} {90}^{\circ}$ triangle are:

$1 : \sqrt{3} : 2$

I hope that this was helpful.

• This key question hasn't been answered yet.

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