One statement relating the lengths of the sides in a right triangle is provided by Pythagoras' theorem. A commonly-used formulation of the theorem is given here.
Pythagoras' Theorem 5.3.3
Consider a right triangle with the right angle at vertex .
The sum of the areas of the squares on the legs a and b equals the area of the square on the hypotenuse c. This statement can be written as an equation (see also the triangle in the figure):
If the sides of the triangle are denoted in another way, the equation has to be adapted accordingly!
Example 5.3.4
Suppose we have a right triangle with legs (short sides) of length and . The length of the hypotenuse can be calculated by means of Pythagoras' theorem:
Exercise 5.3.5
Consider a right triangle with the right angle at vertex , hypotenuse , and altitude (height) . The line segment has the length . Here, is the perpendicular foot of the altitude . Calculate the length of the two legs and .
We apply Pythagoras' theorem to the triangle that has a right angle at the vertex . Then, we have
Now, we apply Pythagoras' theorem to the given right triangle :
Thales' theorem is another important theorem that makes a statement on right triangles.
Thales' Theorem 5.3.6
If the triangle has a right angle at the vertex , then vertex lies on a circle with radius whose diameter is the hypotenuse .
The converse statement is also true. Construct a half-circle above a line segment . If the points and are joint to an arbitrary point on the half-circle, then the resulting triangle is always right-angled.
Example 5.3.7
Construct a right triangle with a given hypotenuse and altitude .
First, draw the hypotenuse
Let the middle of the hypotenuse be the centre of a circle with radius .
Then draw a parallel to the hypotenuse at distance . This parallel intersects Thales' circle in two points and .
Together with the points and , each of these intersections points forms a triangle possessing the required properties, i.e. two solutions exist. Two further solutions are obtained if the construction is repeated drawing a second parallel below the hypotenuse. The constructed triangles are different in position but concerning shape and size these triangles are "congruent" (see also Section 5.3.13).
Exercise 5.3.8
Find the maximum altitude (height) of a right triangle with hypotenuse .
The maximum altitude is the radius of the Thales circle on the hypotenuse. Hence, .
This material is above course level and is not required for the exercises and tests.
In a right triangle, some statements beyond Pythagoras' theorem hold. To study them, we will use the notation illustrated below:
Consider a right triangle with the right angle at the vertex . The altitude intersects the hypotenuse of the triangle in the point , called the perpendicular foot. Furthermore, let and .
Right Triangle Altitude Theorem 5.3.9
The area of the square on the altitude equals the area of the rectangle created by the two hypotenuse segments:
Cathetus Theorem 5.3.10
The area of the square on a leg (cathetus) equals the area of the rectangle created by the hypotenuse and the hypotenuse segment adjacent to the leg:
Example 5.3.11
Let a right triangle with the legs and be given. The length of the hypotenuse can be calculated by means of Pythagoras' theorem:
According to the cathetus theorem the hypotenuse segments and are:
According to the altitude theorem the altitude is:
Exercise 5.3.12
Find the length of the two legs of a given right triangle with hypotenuse , altitude , and hypotenuse segment .
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