#### Chapter 5 Geometry

Section 5.3 All about Triangles

# 5.3.3 Pythagoras' Theorem

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 $C$. 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):

${a}^{2}+{b}^{2}={c}^{2} .$

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 $a=6$ and $b=8$.
The length of the hypotenuse can be calculated by means of Pythagoras' theorem:

$c=\sqrt{{c}^{2}}=\sqrt{{a}^{2}+{b}^{2}}=\sqrt{36+64}=\sqrt{100}=10 .$

##### Exercise 5.3.5
Consider a right triangle $ABC$ with the right angle at vertex $C$, hypotenuse $c=\frac{25}{3}$, and altitude (height) ${h}_{c}=4$. The line segment $\stackrel{‾}{DB}$ has the length $q=\left[\stackrel{‾}{DB}\right]=3$. Here, $D$ is the perpendicular foot of the altitude ${h}_{c}$. Calculate the length of the two legs $a$ and $b$.
Thales' theorem is another important theorem that makes a statement on right triangles.
##### Thales' Theorem 5.3.6 If the triangle $\mathrm{ABC}$ has a right angle at the vertex $C$, then vertex $C$ lies on a circle with radius $r$ whose diameter $2r$ is the hypotenuse $\stackrel{‾}{AB}$.

The converse statement is also true. Construct a half-circle above a line segment $\stackrel{‾}{AB}$. If the points $A$ and $B$ are joint to an arbitrary point $C$ on the half-circle, then the resulting triangle $\mathrm{ABC}$ is always right-angled.
##### Example 5.3.7
Construct a right triangle with a given hypotenuse $c=6 \mathrm{cm}$ and altitude ${h}_{c}=2.5 \mathrm{cm}$.
 First, draw the hypotenuse $c=\stackrel{‾}{AB} .$ Let the middle of the hypotenuse be the centre of a circle with radius $r=c/2$. Then draw a parallel to the hypotenuse at distance ${h}_{c}$. This parallel intersects Thales' circle in two points $C$ and $C\text{'}$. Together with the points $A$ and $B$, 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) ${h}_{c}$ of a right triangle with hypotenuse $c$.