Right triangle

Triangles are polygons that are made up of three sides, so we can say that the triangle is a flat figure that is made up of three different segments .

What is the right triangle?

The right triangle is the triangle that has a right angle which has a measure of 90 degrees and two angles that are acute , which means that they measure less than ninety degrees.

• Characteristics of the right triangle
• Properties
• Types of right triangle
• Elements
• Height
• Area
• Perimeter
• Trigonometric ratios
• How to calculate the angle of a right triangle
• Examples

Characteristics of the right triangle

The most important characteristics of right triangles are:

• They are polygons that have three sides.
• They are made up of three segments .
• They all have a right angle of ninety degrees.
• The two angles that are not right will always be acute angles .
• The right angle is made up of the two shorter sides .
• They can be isosceles or scalene triangles .

Properties

Right triangles have several properties mentioned below:

• They all have two acute angles and their hypotenuse is larger than the legs.
• The square of the hypotenuse is equal to the sum of the square of the legs .
• The sum of the hypotenuse and the diameter of a circle inscribed in the triangle is equal to the sum of the legs .
• When we calculate the area, one leg can be considered as the base and the other leg as the height .
• The median of the hypotenuse decomposes a scalene right triangle into two triangles.
• The median hypotenuse of an isosceles right triangle decomposes it into two congruent and equivalent isosceles right triangles .
• Two right triangles that have a common hypotenuse , and the right angles in opposite half planes determined by the line containing the hypotenuse, form a bi-rectangle quadrilateral .
• The median starting from the right angle is equal to half the hypotenuse .

Types of right triangle

There are two different types of right triangles, these are:

• Isosceles right triangle : it is the triangle that has a 90º right angle and two 45º angles. The two legs are the same.
• Scalene right triangle : it is the right triangle that has all the different angles always having one of them of 90º. The sides are also different.

Elements

The elements of the right triangle are:

• Legs : they are the sides of the triangle that together form the right angle .
• Hypotenuse : it is the largest side having the triangle opposite the angle right .
• Right angle : it is a 90º angle that is formed by the two legs .
• Acute angles : are the other two angles of the triangle (α and β) less than 90º. The sum of the two acute angles is 90º.

Height

The height of a right triangle can be found using the following theorems:

Height theorem : In any right triangle, the height relative to the hypotenuse is the geometric mean between the orthogonal projections of the legs on the hypotenuse.

h / m = m / h

If we multiply the two members of the equality by hn then we can obtain: h 2 = mn, so that h = √ (mn)

Area

The area that has a right triangle will always have a right angle of 90 °, so its height must coincide with one of its sides (a). The area of ​​a right triangle is half the product of the two sides that form the right angle (legs a and b). The formula to calculate the area of ​​the triangle is as follows:

Area = (b a) / 2

Where b is the base and the side that coincides with the height.

Perimeter

We know how the perimeter of a right triangle is the sum of the three sides . To find out, we apply a simple formula, which is the following:

Perimeter = a + b + c

Where a, b and c represent the measures of the triangle.

In addition, the right triangle also complies with the Pythagorean theorem , so the hypotenuse (c) can be expressed from the legs (a and b). The formula for this type of operation is as follows:

Perimeter = a + b √ (a² + b²)

Where a and b are the legs that form the right angle.

Trigonometric ratios

Right triangles have different trigonometric ratios which help us to know the relationship between the sides and the angles of the triangle. Its main function is to show us how much the internal angles of the triangle measure when we know the lengths of two sides of the triangle. It is important to remember that right triangles always have a 90 ° angle . There are three common trigonometric ratios which are:

• Sine (sin): is the ratio that exists between the opposite leg and the hypotenuse and each of them corresponds to an angle.
• Cosine (cos): is the result of dividing the adjacent side by the hypotenuse.
• Tangent (tan): result of dividing the lengths of the opposite and adjacent sides at angle α.

Therefore, we have to:

• Opposite Side / Hypotenuse = Sine of the angle
• Adjacent Side / Hypotenuse = Cosine of Angle
• Opposite Side / Adjacent Side = Tangent of the angle

How to calculate the angle of a right triangle

The heights of the right triangles are associated with the legs (a and b). Therefore, ha = b and hb = a . The height associated with the hypotenuse is known as hc. The three heights of the triangle converge at the orthocenter , H at the vertex C of the right angle.

In order to calculate the height associated with side c (the hypotenuse) , the height theorem is used .

The height h can be obtained knowing the three sides of the right triangle and the following formula is applied:

H = (a b) /  c

The right triangle has a right angle of 90 °, so its height agrees with one of its sides (a) . Its area will then be half the product of the two sides that form the right angle (legs a and b) . The following formula is used:

Area = (b a) /  2

Examples

Some examples of how to solve the areas of right triangles are as follows:

Let be a right triangle with the sides that form the right angle (a and b) known, where a = 3 cm and b = 4 cm. What is its area?

We apply the previous formula for the area of ​​a right triangle (b · a) / 2 and it  is obtained that its area is 6 cm² .

The area of ​​a right triangle can be obtained from the hypotenuse and the height of the triangle associated with it using the height theorem.

Let n and m be the projections of the legs (bya). Then the area of ​​a right triangle is defined by the following formula:

Area = (c √ (n m)) / 2

Formula for the area of ​​a right triangle by the height theorem. This method is useful if the legs (a and b) are not known.