ARE THERE SCALENE TRIANGLES WITH A RIGHT ANGLE? - MATHS - 2025

There are many scalene triangles with a right angle. Before moving on to the topic, it is first necessary to 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 90º each.

- Rectangle: are those triangles that have a right angle, that is, an angle that measures 90º, 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 90º.

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. It only remains 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

According to the length of their sides, the triangles are classified into:

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

- Isosceles: are the 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

A question equivalent to the one in the title is "Are there triangles that have three sides with different measures and this one has an angle of 90º?"

The answer as said 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 we can see that the length of 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 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 has been chosen.

Examples

-If we consider a right triangle whose legs have lengths of 3 and 4 respectively, then by Pythagoras' 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 we conclude that ABC is a scalene right triangle.

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

References

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4. Mitchell, C. (1999). Dazzling Math Line Designs. Scholastic Inc.
5. R., MP (2005). I draw 6th. Progress.
6. Ruiz, Á., & Barrantes, H. (2006). Geometries. Editorial Tecnologica de CR.