7 junio, 2024

Acute triangle: concept, characteristics, types, examples

What are acute triangles?

The acute triangles are those whose three internal angles are acute angles; that is, the measure of each of those angles is less than 90 degrees. Not having any right angle, we have that the Pythagorean theorem does not hold for this geometric figure.

Therefore, if we want to have some type of information about any of its sides or angles, it is necessary to make use of other theorems that allow us to access said data. The ones that we can use are the law of sines and the law of cosines.

Characteristics of an acute triangle

Among the characteristics that this geometric figure has, we can highlight those that are given by the simple fact of being a triangle. Among these we have:

– A triangle is a polygon that has three sides and three angles.

– The sum of its three internal angles is equal to 180°.

– The sum of two of its sides is always greater than the third.

As an example let’s look at the following triangle ABC. In general, we identify its sides with a lower case letter and its angles with a capital letter, in such a way that one side and its opposite angle have the same letter.

From the characteristics already given, we know that:

A + B + C = 180°

a + b > c, a + c > b and b + c > a

The main characteristic that distinguishes this type of triangle from the rest is that, as we already mentioned, its internal angles are acute; that is, the measure of each of its angles is less than 90°.

Acute triangles, together with obtuse triangles (those in which one of its angles has a measure greater than 90°), are part of the set of oblique triangles. This set is formed by the triangles that are not rectangles.

As oblique triangles are part of it, in order to solve problems involving acute triangles, we must make use of the sine theorem and the cosine theorem.

sines theorem

The law of sines tells us that the ratio of a side to the sine of its opposite angle is equal to twice the radius of the circle formed by the three vertices of said triangle. That is to say:

2r= a/sin(A)= b/sin(B)= c/sin(C)

cosine theorem

On the other hand, the cosine theorem gives us these three equalities for any triangle ABC:

a2= b2 + c2 -2bc*cos(A)

b2= a2 + c2 -2ac*cos(B)

c2= a2 + b2 -2ab*cos(C)

These theorems are also known as the law of sine and the law of cosine, respectively.

Another characteristic that we can give of acute triangles is that two of these are equal if they meet any of the following criteria:

If all three sides are equal.
If they have one side and two angles equal to each other.
If they have two equal sides and one angle.

Types of acute triangles

We can classify acute triangles based on their sides. These might be:

Equilateral acute triangles

They are acute triangles that have all their sides equal and, therefore, all their internal angles have the same value, which is A = B = C = 60° degrees.

As an example let’s take the following triangle, whose sides a, b and c have a value of 4.

Isosceles acute triangles

These triangles, in addition to having acute internal angles, have the characteristic of having two of their equal sides and the third, which is generally taken as the base, different.

An example of this type of triangle can be one whose base is 3 and its other two sides have a value of 5. With these measurements, it would have the opposite angles to the equal sides with the value of 72.55° and the opposite angle of the base would be 34.9°.

Scalene acute triangles

These are the triangles that have all their sides different two by two. Therefore, all its angles, in addition to being less than 90°, are different two by two.

Triangle DEF (whose measures are d = 4, e = 5 and f = 6 and whose angles are D = 41.41°, E = 55.79° and F = 82.8°) is a good example of an acute triangle. scalene.

Resolution of acute triangles

As we said before, to solve problems involving acute triangles, it is necessary to use the sine and cosine theorems.

Example 1

Given a triangle ABC with angles A = 30°, B = 70° and side a = 5cm, we want to know the value of angle C and sides b and c.

The first thing we do is use the fact that the sum of the interior angles of a triangle is 180°, in order to obtain the value of angle C.

180°= A + B + C = 30°+70° + C = 100° + C

We clear C and we are left with:

C = 180° – 100° = 80°

Since we already know the three angles and one side, we can use the Law of Sines to determine the value of the remaining sides. By the theorem we have that:

a/sin(A) = b/sin(B) and a/sin(A)= c/(sin(C)

We clear b from the equation and we are left with:

b = (a*sin(B))/sin(A) ≈ (5*0.940) / (0.5) ≈ 9.4

Now all that remains is to calculate the value of c. We proceed in an analogous way as in the previous case:

c = (a*sin(C))/sin(A) ≈ (5*0.984)/(0.5) ≈ 9.84

Thus we obtain all the data of the triangle. As we can see, this triangle falls into the category of scalene acute triangle.

Example 2

Given a triangle DEF with sides d = 4cm, e = 5cm and f = 6cm, we want to know the value of the angles of said triangle.

For this case we will use the law of cosines, which tells us that:

d2= e2 + f2 – 2efcos(D)

From this equation we can solve cos(D), which gives us as a result:

Cos(D)=((4)2 – (5)2 –(6)2)/(-2*5*6) =0.75

From here we have that D≈ 41.41°

Using now the sinom theorem we have the following equation:

d/(sin(D)= e/(sin(E)

Solving for sin(E), we have:

sin(E)= e*sin(D)/d = (5*0.66)/4 ≈ 0.827

From here we have that E≈55.79°

Finally, using that the sum of the interior angles of a triangle is 180°, we have that F≈82.8°.

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