Atmetric relationshipsare equations that relate the measurements of the sides and some other segments on one right triangle. To define these relationships, it is important to know these segments.
Rectangle Triangle Elements
The following figure is a trianglerectangle ABC, whose right angle is  and is cut by height AD:
In this triangle, note that:
The letter The is the measure of hypotenuse;
The letters B and ç are the measurements of the collared peccaries;
The letter H is the measure of height of the right triangle;
The letter no and the projection of the AC leg over the hypotenuse;
The letter m and the projection of the BA leg over the hypotenuse.
Pythagorean Theorem: first metric relation
O Pythagorean theorem is the following: the square of the hypotenuse is equal to the sum of the squares of the legs. It is valid for all trianglesrectangles and can be written as follows:
The2 = b2 + c2
*a is hypotenuse, b and c are peccaries.
Example:
What is the diagonal measurement of a rectangle whose long side is 20 cm and the short side is 10 cm?
Solution:
THE diagonal of a rectangle divides it into two right triangles. This diagonal is the hypotenuse, as shown in the following figure:
To calculate the measure of this diagonal, just use the theoreminPythagoras:
The2 = b2 + c2
The2 = 202 + 102
The2 = 400 + 100
The2 = 500
a = √500
a = approximately 22.36 cm.
second metric relation
THE hypotenuse of trianglerectangle is equal to the sum of the projections of their legs on the hypotenuse, that is:
a = m + n
third metric relation
O square gives hypotenuse on one trianglerectangle it is equal to the product of the projections of their legs on the hypotenuse. Mathematically:
H2 = m·n
Thus, if it is necessary to find the measure of the hypotenuse knowing only the measures of the projections, we can use this metric relationship.
Example:
A triangle whose projections of the cats on the hypotenuse measure 10 and 40 centimeters how tall are they?
H2 = m·n
H2 = 10·40
H2 = 400
h = √400
h = 20 centimeters.
fourth metric relation
It is used to find the measurement of a collared when the measurements of your projection about the hypotenuse and the own hypotenuse are known:
ç2 = an
and
B2 = an
realize that B is the measure of the AC collar, and no it is the measure of your projection onto the hypotenuse. The same goes for ç.
Example:
Knowing that the hypotenuse on one trianglerectangle measures 16 centimeters and that one of your projections measures 4 centimeters, calculate the measure of the leg adjacent to this projection.
Solution:
The side adjacent to a projection can be found from any of these relationsmetrics: ç2 = am or b2 = an, as the example does not specify the collared in question. Thus:
ç2 = a·m
ç2 = 16·4
ç2 = 64
c = √64
c = 8 centimeters.
fifth metric ratio
The product between the hypotenuse(The) and the height(H) of a right triangle is always equal to the product of the measurements of its legs.
oh = bc
Example:
what is the area of a trianglerectangle whose sides have the following measurements: 10, 8 and 6 centimeters?
Solution:
10 centimeters is the measurement on the longest side, so this is the hypotenuse and the other two are peccaries. To find the area, you need to know the height, so we'll use this metric relationship to find the height of this triangle and then we will calculate your area.
a·h = b·c
10·h = 8·6
10·h = 48
h = 48
10
h = 4.8 centimeters.
A = 10·4,8
2
A = 48
2
H = 24 cm2
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-sao-relacoes-metricas-no-triangulo-retangulo.htm