Solving Special Right Triangles: 30°-60°-90° and 45°-45°-90°

For sure you have read about special right triangles, and how it is stated that the side opposite to the 30° angle helps to define the other two sides in a 30°-60°-90°, and the legs in a 45°-45°-90° are used to define the hypotenuse; but you still don't know why this is true!

You will learn in this lesson why, and how you may remember it in the future. We start with an equilateral triangle to find the 30°-60°-90° side relationship, and we continue with a square to find the 45°-45°-90° side relationship; this way you will interiorize in your mind the concepts and they will come to you whenever you need them again in a problem solving process!

Lesson's Content

Lesson's Glossary

Acute triangle
A triangle whose angles are acute.

Altitude
Height

Altitude of a triangle
The perpendicular segment from a vertex to the line containing the opposite side of a triangle.

Angle bisector
A ray that is in the interior of an angle and forms two equal angles with the sides of that angle.

CPCTC

Corresponding Parts of Congruent Triangles are Congruent.

Equilateral triangle
A triangle whose sides are equal in length.

Isosceles triangle
A triangle with two sides of equal length.

Median
The segment connecting the vertex of an angle in a triangle to the midpoint of the side opposite it.

Obtuse triangle
A triangle with one acute angle.

Perpendicular bisector
The bisector of a segment perpendicular to it.

Right triangle
A triangle that has a 90 degree angle.

Scalene triangle
A triangle with no equilateral sides.

Triangle
A polygon with three sides.

Interactive Geometric Applets: Relevant Theorems.

In the lesson we explained how to find the angle side relationships in special right triangles.

For this you will need to know what are the line segments that we may draw in a triangle, and

that connect side with angle, or divide a side, and/or an angle. This applet presents you

these special segments.

Drag any vertex in the triangles to verify how is that they meet their definition.

A special right triangle 30°-60°-90° has a longer leg that is square root of three times the shorter leg,

and a hypotenuse that is two times the shorter leg. This geometric interactive applet will allow

to drag the slider and see several instances of this type of special right triangle.

In a special right triangle 45°-45°-90° the legs are congruent, and the hypotenuse is

given by the square root of two times any of the legs. This is an isosceles right triangle.

Play with the slider to generate several instances of this 45°-45°-90° triangle

Vocabulary Puzzle Interactive

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