ACUTE TRIANGLE: CHARACTERISTICS AND TYPES - MATHS - 2024

The acute triangles are those whose three internal angles are acute angles; that is, the measure of each of these angles is less than 90 ° degrees. By 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 have access to said data. The ones we can use are the sine theorem and the cosine theorem.

characteristics

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 a general way, we identify its sides with a lowercase 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 feature 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 their angles has a measure greater than 90 °), are part of the set of oblique triangles. This set is made up of the triangles that are not right angles.

As the oblique triangles are part, we have to be able to solve problems involving acute triangles, we must make use of the sine theorem and the cosine theorem.

Sine theorem

The sine theorem tells us that the ratio of one 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:

a ² = b ² + c ² -2bc * cos (A)

b ² = a ² + c ² -2ac * cos (B)

c ² = a ² + b ² -2ab * cos (C)

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

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

- If they have the same three sides.

- If they have one side and two equal angles to each other.

- If they have two equal sides and an angle.

Types

Acute triangles can be classified according to their sides. These might be:

Equilateral acute triangles

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

The triangle DEF (whose measures are d = 4, e = 5 and f = 6 and its 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 the 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 have:

C = 180 ° - 100 ° = 80 °

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

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

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

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

Now we only need to calculate the value of c. We proceed in the same 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 cosine law, which tells us that:

d ² = e ² + f ² - 2efcos (D)

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

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

Hence we have D≈ 41.41 °

Using now the senom 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

Hence we have E≈55.79 °

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

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