Name each segment

In a right triangle, the side opposite the right angle is the hypotenuse. Call her $a$ and the boys $b,c$. The other angles are acute and add up to $90°$.

Draw the height $h$ to the hypotenuse. It divides $a$ into two projections: $m$, associated with the side $b$, and $n$, associated with the side $c$. Thus, $m+n=a$.

Demonstration by similarity

The two smaller triangles have a right angle and an acute angle the same as the original triangle. They are similar to him. Comparing corresponding sides:

$$\frac b a=\frac m b\Rightarrow b^2=am,\qquad \frac c a=\frac n c\Rightarrow c^2=an.$$

Adding the two equalities and using $m+n=a$:

$$b^2+c^2=a(m+n)=a^2.$$

We obtained the Pythagorean theorem. It was deduced from relations between similar sides, without depending on an example of particular measurements.

The height by two paths

The area can be calculated by the legs or the hypotenuse and its height:

$$\frac{bc}{2}=\frac{ah}{2}\Rightarrow ah=bc.$$

Squaring and substituting $b^2=am$ and $c^2=an$:

$$h^2=\frac{b^2c^2}{a^2}=\frac{am\cdot an}{a^2}=mn.$$

Therefore, $h=\sqrt{mn}$. All of these measures are positive.

The median to the hypotenuse

The midpoint M of the hypotenuse is the same distance from the three vertices. To see why, complete the triangle until it forms a rectangle. The diagonals of the rectangle are equal and cut in half. Each half measures $a/2$.

Thus, the median drawn from the right angle to M measures $a/2$. This also shows that the circle with center M and radius $a/2$ passes through the three vertices.

Worked example

For legs 6 and 8 cm, $a=\sqrt{36+64}=10$ cm. Then:

$$h=\frac{6\cdot8}{10}=4{,}8\text{ cm},$$

$$m=\frac{6^2}{10}=3{,}6\text{ cm},\qquad n=\frac{8^2}{10}=6{,}4\text{ cm}.$$

The sum of the projections is 10 cm and the median measures 5 cm, confirming the relationships.

In everyday life

  • Bricklayer’s square. By marking 3 m, 4 m and checking 5 m diagonal, the bricklayer guarantees a straight corner.
  • TV size. TV inches measure the diagonal: width and height are the sides.
  • Shortcut. Cutting a path diagonally across a plot of land measuring 30 m by 40 m saves 20 m.
The bricklayer’s square: 3 m on one wall, 4 m on the other; if the diagonal gives 5 m, the corner is straight, because $3^2 + 4^2 = 5^2$.

Exercises with resolution

1. The height divides the hypotenuse into 4 and 9 cm. Calculate the hypotenuse and height.

Check resolution 1

$a=4+9=13$ cm and $h=\sqrt{4\cdot9}=6$ cm.

2. The hypotenuse measures 26 cm. How long is the median that starts from the right angle?

Check resolution 2

Half of the hypotenuse: $26/2=13$ cm.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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