In this lesson, you will learn about the properties of and theorems associated with right triangles, which have a wide range of applications in math and science. Specifically, we will discuss and prove the Pythagorean theorem and the right triangle altitude theorem. Let’s get started.
What Are Right Triangles?
Triangles come in many different sizes and orientations, but there’s one type of triangle that stands out in particular: the right triangle. Right triangles are used in countless applications, from designing fighter jets to proving complex mathematical theorems and beyond. So what are these seemingly omnipotent geometrical structures?
A right triangle is a triangle with one of its angles measuring 90 degrees. Also, by definition, an angle measuring 90 degrees is a right angle and should be denoted by a little square, as we will see soon enough. Let’s look at some examples of right triangles. Note that each of the displayed triangles has a little square to designate the right angle. This is very important since some triangles are not considered to be a right triangle because angle C is not designated as a right angle as in this image:
An important property of right triangles is that the measures of the non-right angles (denoted alpha and beta in this figure) must add up to 90 degrees. This stems from the fact that the sum of all angles in a triangle is 180 degrees, so alpha plus beta plus 90 equals 180 degrees. Implying that, alpha plus beta equals 90 degrees.
By the way, angles that add up to 90 degrees are also called complementary angles, in case you read that somewhere else. The sides opposite angles alpha and beta are called legs, while the side opposite the right angle is called the hypotenuse.
With all of that out of the way, you’re now ready to proceed to two important theorems that have made right triangles super famous.
The Pythagorean Theorem
The Pythagorean theorem is named after the Greek mathematician Pythagoras, although it was also discovered independently by other ancient civilizations. It describes the relation between the two legs and hypotenuse of a right triangle. Take a look at the figure below. Note that the two legs are denoted as a and b, while the hypotenuse is denoted as c. The Pythagorean theorem states that the same of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse, or more simply, a squared plus b squared equals c squared.
Let’s look at a proof of this theorem proposed by James Garfield. We start with a right triangle, duplicate it, and place it next to the original copy as shown.
Adding an extra line from the tip of angle alpha to the tip of angle beta, as shown by the orange line, we turn the whole figure into a trapezoid. Note that the angle opposite line segment k must measure 90 degrees. The formula for the area of a trapezoid is one-half height times sum of the bases. Looking at our figure, both the sum of the bases and the height are equal to a plus b, so the area of our trapezoid is equal to one-half times a plus b times a plus b.
Now, summing up the areas of the two duplicate triangles with the third triangle formed by line segment k and recalling that the area of a triangle is one-half times base times height, we have this:
Right Triangle Altitude Theorem
With that out of the way, let’s move on to the another important theorem. The right triangle altitude theorem states that in a right triangle, the altitude drawn to the hypotenuse forms two right triangles that are similar to each other as well as to the original triangle. Starting with triangle ABC, drop a perpendicular from angle C onto the hypotenuse as shown here:
This perpendicular is also know as the altitude of the triangle. Now, we label the newly formed angles as shown here. Let’s see how we can prove the right angle altitude theorem from this figure.
Note the following pairs of complementary angles. Therefore, since angle alpha in triangle ADC is congruent to angle alpha in triangle ACB, the angles ADC and ACB are congruent. Triangles ADC and ACB are similar to each other. In addition, since angle beta in triangle ACD is congruent to angle beta in triangle ACB, and angles CDB and ACB are congruent, triangles CDB and ACB are similar to each other.
If two triangles are similar to the same triangle, then they must be similar to each other. Since triangles ADC and CDB are both similar to triangle ACB, triangle ADC is similar to triangle CDB.
Let’s summarize everything we’ve learned. A right triangle is a triangle with one of its angles measuring 90 degrees. Also, an angle measuring 90 degrees is a right angle. Right angles must be donated by a little square in geometric figures. If we neglect to designate a right angle this way, it cannot be assumed to be a right angle, even if it visually appears to be about 90 degrees.
Complementary angles are angles that add up to 90 degrees. You have also seen a proof of the Pythagorean theorem, which states that the same of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Another theorem we’ve covered, the right triangle altitude theorem, states that in a right triangle, the altitude drawn to the hypotenuse forms two right triangles that are similar to each other as well as to the original triangle. You are now an expert on right triangles.