By Allen Ma, Amber Kuang

In geometry, special right triangles are great to work with because the ratio of their sides will always be the same, making calculations easier. The two special triangles you need to know are the isosceles (or 45-45-90) and 30-60-90 right triangles.

You can use your knowledge of special right triangles to answer the following questions.

Practice questions

  1. The length of the altitude of an equilateral triangle is

    image0.png

    Find the length of a side of the triangle.

  2. In isosceles right

    image1.png

    Rectangle PLCE,

    image2.png

    with diagonal

    image3.png

    drawn forms a 30-degree angle at

    image4.png

    Find the length of

    image5.png

Answers and explanations

  1. 8

    An equilateral triangle is a triangle with three congruent sides and three congruent angles. Each angle therefore measures 60 degrees. When you draw the altitude, it creates two

    image6.png

    as in the following figure.

    image7.png

    The pattern of the sides of a

    image8.png

    is as follows:

    image9.png

    The altitude is the side opposite the 60-degree angle; therefore,

    image10.png

    A side of the equilateral triangle is opposite the 90-degree angle, so it’s equal to 2x; therefore, a side of the triangle equals

    image11.png

  2. image12.png

    The pattern of the sides of an isosceles right triangle

    image13.png

    is as follows:

    image14.png

    In this problem, the side opposite the 45-degree angle is

    image15.png

    therefore,

    image16.png

    As shown in the figure,

    image17.png

    diagonal

    image18.png

    divides this rectangle into two

    image19.png

    The pattern of the sides of a

    image20.png

    is as follows:

    image21.png

    Line segment PE, measuring 24, is the side opposite the 60-degree angle, so it’s equal to

    image22.png

    therefore,

    image23.png

    Line segment PL is the side opposite the 60-degree angle, so it equals x. Therefore,

    image24.png