Special Right Triangles

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Trigonometry: Special Right Triangles


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.

  • Answer:

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


    #45-45-90" "# refers to the angles of the triangle.

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

    There are #color(blue)("two equal angles")#, so this is an isosceles triangle.

    It therefore also has #color(blue)(" two equal sides.")#

    The third angle is #90°#. It is a #color(blue)("right-angled triangle")# therefore Pythagoras' Theorem can be used.

    The #color(blue)("sides are in the ratio " 1 :1: sqrt2)#

    It has #color(blue)("one line of symmetry")# - the perpendicular bisector of the base (the hypotenuse) passes through the vertex, (the #90°# angle).

    It has #color(blue)("no rotational symmetry.")#

  • #mathbf{30^circ"-"60^circ"-"90^circ}# Triangle

    The ratios of three sides of a #30^circ"-"60^circ"-"90^circ# triangle are:


    I hope that this was helpful.

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