7 junio, 2024

Are there scalene triangles with a right angle?

There are many scalene triangles with a right angle. Before advancing on the subject, it is necessary to first know the different types of triangles that exist. Triangles are classified by two classes, which are: their internal angles and the lengths of their sides.

The sum of the internal angles of any triangle is always equal to 180º. But according to the measures of the internal angles they are classified as:

acute angle: are those triangles such that their three angles are acute, that is, they measure less than 90° each.

Rectangle: are those triangles that have a right angle, that is, an angle that measures 90°, and, therefore, the other two angles are acute.

obtuse angle: are the triangles that have an obtuse angle, that is, an angle whose measure is greater than 90°.

Scalene triangles with a right angle

The interest in this part is to determine if a scalene triangle can have a right angle.

As stated above, a right angle is an angle whose measure is 90º. All that remains is to know the definition of a scalene triangle, which depends on the length of the sides of a triangle.

Classification of triangles according to their sides

Triangles are classified according to the length of their sides:

Equilateral: are all those triangles such that the lengths of its three sides are equal.

IsoscelesThey are triangles that have exactly two sides of equal length.

Scalene: are those triangles in which the three sides have different measures.

Formulation of an equivalent question

An equivalent question to the one in the title is “Are there triangles that have three sides with different measures and this has an angle of 90°?”

The answer as stated at the beginning is Yes. It is not very difficult to justify this answer.

If you look carefully, no right triangle is equilateral, this can be justified thanks to the Pythagorean theorem for right triangles, which says:

Given a right triangle such that the lengths of its legs are “a” and “b”, and the length of its hypotenuse is “c”, we have that c²=a²+b², with which it can be seen that the length of the hypotenuse “c” is always greater than the length of each leg.

Since nothing is said about «a» and «b», then this implies that a right triangle can be either Isosceles or Scalene.

Then, it is enough to choose any right triangle such that its legs have different measures, and thus a scalene triangle that has a right angle will have been chosen.

examples

-If a right triangle is considered whose legs have lengths of 3 and 4 respectively, then by the Pythagorean theorem it can be concluded that the hypotenuse will have a length of 5. This implies that the triangle is scalene and has a right angle.

-Let ABC be a right triangle with legs of measures 1 and 2. Then the length of its hypotenuse is √5, with which it is concluded that ABC is a scalene right triangle.

Not every scalene triangle has a right angle. A triangle like the one in the following figure can be considered, which is scalene, but none of its internal angles are right.

Also, not every right triangle is scalene. If we consider a right triangle whose legs both measure 1, then the hypotenuse will have a measure of √2. Therefore, the right triangle is isosceles.

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