10.02 Extension: Special right triangles
The $45^\circ-45^\circ-90^\circ$45°−45°−90° triangle theorem
While the Pythagorean theorem can apply to any kind of right triangle, there are particular types of right triangles whose side lengths and angles have helpful properties.
The first kind of triangle that we want to look at is the $45^\circ-45^\circ-90^\circ$45°−45°−90° triangle.
The first thing to notice about this kind of triangle is that it has two congruent angles and is, therefore, an isosceles triangle. This, in turn, means that the two sides of the triangle that are not the hypotenuse are equal in length.
Knowing this fact means that we can also use the Pythagorean Theorem to find out what the length of the hypotenuse should be. For example, let's look at the case where the two shorter sides are both of length $1$1.
We can then find the length of the hypotenuse, $c$c, in the following way.
| $1^2+1^2$12+12 | $=$= | $c^2$c2 | Using the Pythagorean Theorem |
| $1+1$1+1 | $=$= | $c^2$c2 | Simplifying the powers |
| $c^2$c2 | $=$= | $2$2 | Evaluating the addition |
| $c$c | $=$= | $\sqrt{2}$√2 | Taking the positive square root of both sides |
So, here we have a $45^\circ-45^\circ-90^\circ$45°−45°−90° with all side lengths filled in:
Note that any triangle with angle measures of $45^\circ$45°, $45^\circ$45° and $90^\circ$90° is necessarily similar to this triangle. Using what we know about similar triangles, this means that the sides of any $45^\circ-45^\circ-90^\circ$45°−45°−90° triangle are in the ratio $1:1:\sqrt{2}$1:1:√2.
In any $45^\circ-45^\circ-90^\circ$45°−45°−90° triangle, the hypotenuse is $\sqrt{2}$√2 times as long as either of the other sides of the triangle.
That is, the ratio of side lengths in a $45^\circ-45^\circ-90^\circ$45°−45°−90° triangle is always $1:1:\sqrt{2}$1:1:√2.
Practice questions
Question 1
Consider the triangle below.
Find the exact value of $a$a.
Find the exact value of $c$c.
Question 2
Consider the triangle below.
Find the exact value of $a$a.
Find the value of $b$b.
Question 3
Consider the triangle below.
Find the exact value of $b$b.
Find the exact value of $c$c.
The $30^\circ-60^\circ-90^\circ$30°−60°−90° triangle theorem
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 labeled $x$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.
Practice questions
Question 4
Consider the right triangle below.
Use the Pythagorean Theorem to find the missing side length $b$b.
Using the fact that the sum of the interior angle measures of a triangle is $180^\circ$180°, find the value of $\theta$θ.
Question 5
Consider the triangle below.
Determine the value of $c$c.
Determine the value of $b$b.
Question 6
Consider the triangle below.
A right triangle has a right angle at its base, as indicated by a small square. The angle of one of the vertices has a measure of $60^\circ$60°. The base of the triangle is opposite the $60^\circ$60° angle, and is labeled as "$4$4", indicating a length of $4$4 units. The hypotenuse, which is the side opposite the right angle, is labeled as "$c$c," indicating its unknown length. The third side is adjacent the $60^\circ$60° angle, and is labeled as "$a$a," indicating its unknown length.
Determine the value of $a$a, giving your answer with a rational denominator.
Determine the value of $c$c, giving your answer with a rational denominator.