We have looked at similar right triangles in this unit that are created by drawing an altitude to the hypotenuse. We have also explored how right triangle trigonometry can be used to solve for missing sides and angles. This target focuses on two special right triangles.
The triangles created above are called special right triangles. Once you split the square in half by its diagonal, it creates a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, whose name is based on its angle measures. Once you split the equilateral triangle in half by one of its altitudes, it create a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle.
The triangles created above are called special right triangles. Once you split the square in half by its diagonal, it creates a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, whose name is based on its angle measures. Once you split the equilateral triangle in half by one of its altitudes, it create a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle.
The \(30^{\circ}-60^{\circ}-90^{\circ}\) Triangle
Because every \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle has the same corresponding angle measures, we can say that all \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles are similar to one another by \(AA\sim\).
Because every \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle has the same corresponding angle measures, we can say that all \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles are similar to one another by \(AA\sim\).
\(30^{\circ}-60^{\circ}-90^{\circ}\) Triangle
The side lengths in this type of triangle follow a certain pattern, where \(a\) is any real number. The longest side (opposite the \(90^{\circ}\) angle) is always labelled \(2a\), the side opposite the \(60^{\circ}\) angle is labelled \(a\sqrt3\), and the side opposite the \(30^{\circ}\) angle is labelled \(a\). |
Example 1:
In the following triangle, find \(PM\) and \(AM\). Solution: Since this is a right triangle with one side length and one angle measure given, your first instinct would be to use trigonometry. That isn’t wrong and these triangles can be solved using trig! But here’s another way to obtain exact answers. |
Solution 1 (Scale Factor):
You can use the patterns above to solve for the missing side lengths. We will label the side lengths with \(a\) for the short leg, the short leg length times the square root of three or \(a\sqrt{3}\) for the long leg, and two times the short leg for the hypotenuse or \(2a\). Be sure that the \(2a\) is across from the \(90^{\circ}\) angle and that the \(a\) is opposite the \(30^{\circ}\) angle. |
Set up an equation with the side length that is given to you to solve for \(a\), the scale factor. In this case, since you labelled the side with \(5\) cm to be "\(a\)", our scale factor will be \(a=5\):
Note: Sometimes you will need to take a few extra steps to solve for \(a\). Substitute the value of \(a\) in both expressions for the missing side lengths: |
Solution 2 (Proportions):
You can use similarity and set up proportions in order to solve for the missing lengths. Use the triangle where \(a = 1\). Redraw it so that it’s facing the same way as the given triangle: Because you have at least two pairs of congruent corresponding angles, these triangles are similar. We now can set up a proportion to solve for the missing side lengths: |
Solution 1 (Scale Factor):
Solution 2 (Proportions):
The \(45^{\circ}-45^{\circ}-90^{\circ}\) Triangle
Because every \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle has the same corresponding angle measures, we can say that all isosceles right triangles are similar to one another by \(AA\sim\). It also has a special pattern for its side lengths.
Because every \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle has the same corresponding angle measures, we can say that all isosceles right triangles are similar to one another by \(AA\sim\). It also has a special pattern for its side lengths.
\(45^{\circ}-45^{\circ}-90^{\circ}\) Triangle
The side lengths in an isosceles right triangle follow a certain pattern, where \(a\) is any real number. The length of the congruent legs of the triangle are labelled \(a\). The hypotenuse (opposite the \(90^{\circ}\) is labelled square root of two times the length of a leg or \(a\sqrt2\). |
Example 4:
Find the missing side lengths.
Find the missing side lengths.
Rationalizing the denominator:
Now we need to find (\JH\) by substituting the value of \(a\):
Example 6:
Find all the missing side and angle measurements of the two given triangles below:
Find all the missing side and angle measurements of the two given triangles below:
Solution:
Watch the video for the solution.
Watch the video for the solution.
Quick Check
1) Find the missing side lengths using special right triangles.
1) Find the missing side lengths using special right triangles.
2) Find the missing side lengths.
3) Find the length of a diagonal of a square whose perimeter is \(8\) meters.