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PreCalculus
Special Right Triangles
Name _________________________ #_____
Since I am not there with you today, I’ve created this wonderful worksheet of all the things I was going
to tell you. So just imagine my lovely voice saying these words as you read…
Hopefully, this lesson will be somewhat of a review, but it’s okay if it’s not or if you don’t remember.
This is an ______________________ triangle because all sides
are the same length and all angles are 600.
Draw the altitude of the triangle. (that’s the
height and makes a right angle with the
base.)
This forms two right triangles like the one to
the right. Label all angles and side lengths.
Show your work on how to find the height.
Keep this value in simplified radical form
(no decimals)
This is known as the 30-60-90 special right triangle (because those are the angles…did you catch
that?) There is always the same relationship between the side lengths of 30-60-90 triangle.
Let’s call the side across of the 300 angle a. The hypotenuse will always be 2a
and the side across from the 600 angle is 𝑎√3.
Example: Let a = 5 then use the correct trig function to find the other leg.
Show work.
2a
𝑎√3
Check to see if this value is equal to 5√3.
Now use the Pythagorean Theorem to find the hypotenuse. Show work.
a
Did you get 10?
Remembering this relationship with a 30-60-90 triangle will make calculations faster instead of having
to show the Pythagorean Theorem and trig functions to find the missing pieces.
There is also a relationship with a 45-45-90 triangle.
Draw the diagonal of the square. Label the measures
of the angles. And find the hypotenuse (use exact
values…no decimals).
The legs are equal and can be represented with a.
The hypotenuse will always be 𝑎√2 in a 45-45-90
triangle.
Practice: Simplify each radical. (No decimals!)
(1) √20
(2) √90
(3) 2√32
(4) (3√5)
2
Use properties of special right triangles to find the missing side lengths. Answers should be
written in simplified radical form. (No decimals!)
(5)
(6)
(7)
(8)
(9)
(10)