Just like the $45^\circ$45°-$45^\circ$45°-$90^\circ$90° triangle, the $30^\circ$30°-$60^\circ$60°-$90^\circ$90° triangle is another special type of right triangle. We can construct a $30^\circ$30°-$60^\circ$60°-$90^\circ$90° triangle by starting with an equilateral triangle and cutting it into two halves.
Here is an equilateral triangle:
To form a $30^\circ$30°-$60^\circ$60°-$90^\circ$90° triangle, we draw in an altitude from any vertex. In fact, we get two congruent $30^\circ$30°-$60^\circ$60°-$90^\circ$90° triangles by doing so:
In order to look at the relationships between the side lengths of such a triangle, let's suppose that the initial equilateral triangle $\triangle ABC$△ABC had side lengths of $2$2 units each. Since the altitude $\overline{AD}$AD bisects the side $\overline{BC}$BC, this means that the length of the short side $\overline{BD}$BD will be $1$1 unit. Here is this information on the $30^\circ$30°-$60^\circ$60°-$90^\circ$90° triangle $\triangle ABD$△ABD:
There is only one unknown side length left on this triangle, $AD$AD, which has been labelled $x$x on the diagram above. Since $\triangle ABD$△ABD is a right triangle, we can use the Pythagorean Theorem to find this length:
$1^2+x^2$12+x2 | $=$= | $2^2$22 | Using the Pythagorean Theorem |
$1+x^2$1+x2 | $=$= | $4$4 | Simplifying |
$x^2$x2 | $=$= | $3$3 | Subtracting $1$1 from both sides |
$x$x | $=$= | $\sqrt{3}$√3 | Taking the positive square root of both sides |
Here is a $30^\circ$30°-$60^\circ$60°-$90^\circ$90° with all side lengths filled in:
Note that any triangle with angle measures of $30^\circ$30°, $60^\circ$60° and $90^\circ$90° is necessarily similar to this triangle. Using what we know about similar triangles, this means that the sides of any $30^\circ$30°-$60^\circ$60°-$90^\circ$90° triangle are in the ratio $1:\sqrt{3}:2$1:√3:2.
In any $30^\circ$30°-$60^\circ$60°-$90^\circ$90° triangle, the longer leg is $\sqrt{3}$√3 times as long as the shorter leg, and the hypotenuse is $2$2 times as long as the shorter leg.
That is, the ratio of side lengths in a $30^\circ$30°-$60^\circ$60°-$90^\circ$90° triangle is always $1:\sqrt{3}:2$1:√3:2.
Use the Pythagorean Theorem to find the missing side length $c$c.
Consider the triangle below.
Determine the value of $c$c.
Determine the value of $b$b.
Consider the triangle below.
Determine the value of $a$a.
Determine the value of $b$b.