8 junio, 2024

Obtuse triangle: what it is, characteristics, types, examples

We explain what an obtuse triangle is, its elements, characteristics, types, examples and a solved exercise

What is an obtuse triangle?

A obtuse triangle It is the flat, closed figure with three sides, which also contains an obtuse internal angle, that is, greater than 90º and less than 180º.

Any triangle contains 3 internal angles, and if one of them is obtuse, the other two are, by force, acute, since the sum of the internal angles of any triangle is always equal to 180º.

The figure above shows an example of an obtuse triangle, with the lower left internal angle greater than 90º. The remaining internal angles must add up to less than 90º, only then is it true that the sum of the three is equal to 180º.

In addition to obtuse triangles, there are acute triangles, if all their internal angles are acute, and right triangles, when one of the internal angles measures exactly 90º.

Elements of obtuse triangles

Obtuse triangles have the elements common to all triangles: they are flat figures with 3 sides, with 3 internal angles and 3 vertices. In addition, they have notable segments, called cevianssuch as height, median and bisector, and points where the cevians intersect.

Each of these elements is briefly defined as follows:

-Sidesare the segments that make up the figure.

-Verticespoints of intersection of each pair of adjacent sides.

-internal anglesare found between two adjacent sides, on the interior side of the figure, the vertex of the angle coinciding with that of the triangle.

-External angles, are between one side and the extension of the adjacent side, outside the figure, the vertex being common, both of the triangle and of the angle. The sum of the measure between the internal angle and its adjacent external angle is 180º, so they are complementary angles.

-Heightis the measure of the perpendicular segment that joins a vertex with the opposite side, or with its extension.

-Medianline that goes from a vertex to the center of the opposite side.

-Mediatrixsegment perpendicular to one side and passing right through its center.

-Bisectoris a segment that divides in half an interior angle of the triangle.

-orthocenterpoint of intersection of the three heights.

-Barycenteralso called the centroid, is the point where the three medians intersect.

-circumcenterhere the three perpendicular bisectors intersect.

-incenterpoint of confluence of the bisectors.

Once these concepts have been reviewed, some of the most notable characteristics of obtuse triangles are described below.

Characteristics

1.- The sum of the three internal angles of the obtuse triangle is 180º, therefore, only one of its internal angles can be greater than 90º, while the sum of the remaining two is less than 90º.

2.- The longest side of the obtuse triangle is opposite the obtuse angle.

3.- In an obtuse triangle, the heights from the vertices that make an acute angle, intersect with the extensions of the opposite sides.

4.- The orthocenter of an obtuse triangle is outside the figure.

5.- The circumcenter of the obtuse triangle also falls outside the triangle (this is not the case with the acute triangle).

6.- It is only possible to inscribe a square in the obtuse triangle, supporting one of the sides of the square on the longest side of the triangle. Two squares can be drawn, resting the side on the shorter sides of the triangle, leaving an uninscribed vertex (which does not touch the side of the triangle).

7.- Let it be an obtuse triangle with sides (a, b, c), with c being the longest side. The following inequality is valid:

a2+b2 < c2

8.- Let there be two obtuse triangles, whose respective sides are (a, b, c) and (u, v, w). The longest sides of each are c and w, then the following inequality holds:

a∙u + b∙v < c∙w

Types of obtuse triangles

Obtuse triangles can be of two types, according to the length of their sides:

They are briefly described below:

Isosceles triangle

It is the one that has two equal sides and one different one, that is, its sides are (a, a, c).

When the isosceles triangle is both obtuse, the sides of measure «a» are shorter and the side «c» is the longest. The obtuse angle is formed between the equal sides, while the two acute angles are of equal measure and are formed between sides «a» and side «c».

And as it was said in the previous section, the side «c», because it is the longest, is opposite to the obtuse angle.

Scalene triangle

The three sides of the scalene triangle have different measures: (a, b, c).

examples

Example 1

The triangle shown in the following figure is obtuse. The obtuse angle is γ = 114.5º and it is verified that the sum of the three internal angles is 180º:

114.5º + 36.8º + 28.6º ≈ 180º

The longest side is 13.9 units and is opposite the obtuse angle. The aforementioned inequality also holds:

a2+b2 < c2

If a = 7.3 and b = 9.2, then:

7.32 + 9.22 < 13.92

137.93 < 193.2

Example 2

In the Calabi triangle, it is possible to place the largest possible square, in three different ways inside the triangle, as shown in the following figure.

The Calabi triangle is isosceles and obtuse angular. The obtuse angle measures approximately 101.736° and the acute angles at the base both measure 39.13°, also approximately.

solved exercise

The equal sides of an obtuse angled isosceles triangle measure 6 cm, while the longest side measures 10 cm. Calculate the value of the obtuse angle, that of the remaining acute angles and the height from said vertex to the base.

Solution

You can use the law of cosines to find the cosine of the obtuse angle. Then, with the help of the calculator, the angle in question is determined, denoted as γ.

The cosine theorem states that:

c2 = a2 +b2 − 2ab∙cos γ

where γ is the angle between sides a and b. Since the triangle is isosceles, the sides a and b are equal, therefore:

c2 = 2a2 − 2a2∙cos γ

Solving for cos γ:

The obtuse angle is the arc cosine of -0.38889, which is approximately 112.885º. The value of α, the acute angle formed by the 6 cm sides with the 10 cm side is:

2α + 112.885º = 180º

α = (180 – 112.885)/2 = 33.558º

Regarding the height of the triangle, measured from the base, it is obtained by observing that said height divides the triangle into two equal right triangles, with a hypotenuse equal to 6 cm and a base of 5 cm. In such a case, the Pythagorean theorem is applied to directly find the value of h:

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