Features triangular scales, formulas and areas, calculations

scalene triangle This is a three sided polygon, where everyone has a different size or length; for this reason it is given the name scalene, which in Latin means climbing.

Triangles are polygons that are considered the simplest in geometry, because they are formed in three sides, three angles and three vertices. In the case of a scalene triangle, because it has all the different sides, this implies that the three angles will also be different.

Index

  • 1 Characteristics of unequal triangles
    • 1.1 Components
  • 2 Properties
    • 2.1 Internal angle
    • 2.2 Number of sides
    • 2.3 Side inconsistencies
    • 2.4 Inconsistent angles
    • 2.5 Altitude, median, bisector and bisector are not coincidences
    • 2.6 Orthocenter, barycenter, incenter and circumcenter are not concurrent
    • 2.7 Relative height
  • 3 How to calculate the perimeter?
  • 4 How to calculate the area?
  • 5 How to calculate height?
  • 6 How to count sides?
  • 7 Exercises
    • 7.1 First exercise
    • 7.2 The second exercise
    • 7.3 Third exercise
  • 8 References

The characteristics of the triangle are not the same length

A triangle scale is a simple polygon because no side or angle has the same size, unlike an isosceles foot and an equilateral triangle.

Because all sides and angles have different measurements, this triangle is considered an irregular convex polygon.

According to the internal angular amplitude, unequal triangles are classified as:

  • Rectangular triangle scale : all sides are different. One of the angles is straight (90 o ) and the other is sharp and of different sizes.
  • Blunt angle scale of triangle : all sides are different and one of the angles is obtuse (> 90 o ).
  • Sharp triangle triangle scale : all sides are different. All angles are sharp (<90 o ), with different sizes.

Another characteristic of unequal triangles is that because of the mismatch of sides and angles, they do not have an axis of symmetry.

Component

Median : is a line coming out of the midpoint of one side and reaching the opposite point. Three medians agree on a point called centroid or centroid.

The bishop : is a ray which divides each angle into two angles of the same size. The bisector of a triangle agrees at a point called incentro.

Mediatrix : is a segment perpendicular to the side of the triangle, which originates from this center. There are three mediatrices in a triangle and agree on a point called circumcenter.

Height : is the line that moves from the point to the opposite side and also this line is perpendicular to that side. All triangles have three heights that coincide at a point called an orthocenter.

Property

Triangular scales are defined or identified because they have several properties that represent them, derived from the theorems put forward by great mathematicians. They are:

Internal angle

The number of internal angles is always equal to 180 o .

Number of sides

The number of two-sided steps must always be greater than the size of the third side, a + b> c.

The inconsistent side

All sides of an unequal triangle have different sizes or lengths; that is, they are incompatible.

Inconsistent angle

Because all sides of the triangle are not the same, the angles will also be different. However, the number of internal angles will always be equal to 180º, and in some cases, one angle can be blunt or straight, while on the other hand all angles will become acute.

Height, median, bisector and bisector are not coincidences

Like any triangle, scalene has several straight line segments that make it up, such as: height, median, bisector and bisector.

Because of the peculiarities of the sides, in this type of triangle none of these lines will coincide in one.

Orthocenter, barycenter, incenter and circumcenter are not coincidences

Because heights, medians, lines, and lines are represented by different straight line segments, in an unequal triangle, the meeting points – orthocenter, centrocenter, incenter and circumcenter – will be found at different points ( they do not coincide).

Depending on whether the triangle is acute, rectangular or scalent, the orthocenter has a different location:

a. If the triangle is acute, the orthocenter will be inside the triangle.

b. If the triangle is rectangular, the orthocenter will coincide with the straight-side node.

c. If the triangle is blunt, the orthocenter will be outside the triangle.

Relative height

Relative height to the side.

In the case of a scalent triangle, this height will have a different size. Each triangle has three relative heights and to calculate it the formula Stork is used.

How to calculate the perimeter?

Polygon perimeter is calculated by the number of sides.

As in this case, the scalent triangle has all sides of different sizes, the perimeter is:

P = side a + side b + side c.

How to calculate area?

The area of ​​a triangle is always calculated with the same formula, multiplying the base by height and dividing by two:

Area = (base * h) ÷ 2

In some cases the height of the unequal triangle is unknown, but there is a formula proposed by the mathematician Heron, to calculate the area of ​​knowing the three-sided measurements of the triangle.

Where:

  • a, b and c, represent the sides of a triangle.
  • sp, according to the triangular semiperimeter, which is half of the perimeter:

sp = (a + b + c) ÷ 2

If you only have two sides of a triangle and the angle formed between them, the area can be calculated by applying trigonometric ratios. So you have to:

Area = (next to * h) ÷ 2

Where height (h) is the product of one side by the sine of the opposite angle. For example, for each side, the area will:

  • Area = (b * c * cent A) ÷ 2
  • Area = (a * c * cent B) ÷ 2.
  • Area = (a * b * cent C) ÷ 2

How to calculate height?

Because all sides of the scalene triangle are different, it is not possible to calculate the height with the Pythagorean theorem.

From the Heron formula, which is based on measuring three sides of a triangle, the area can be calculated.

The height can be cleared from the general formula area:

The sides are replaced by side measurements a, b or c.

Another way to calculate height when the value of one angle is known is to apply a trigonometric ratio, where height will represent the legs of a triangle ..

For example, when the angle opposite to the height is known, it will be determined by sine:

How to count sides?

When you have the size of two sides and the angle opposite to this, it is possible to determine the third side by applying the cosine theorem.

For example, in triangle AB, height relative to the AC segment is plotted. That way the triangle is divided into two right triangles.

To calculate the c-side (segment AB), the Pythagorean theorem is applied to each triangle:

  • For the blue triangle you must:

2 = h 2 + m 2

As m = b – n, replaced:

2 = h 2 + b 2 (b – n) 2

2 = h 2 + b 2 – 2bn + n 2 .

  • For the pink triangle you must:

2 = a 2 – n 2

It was replaced in the previous equation:

2 = a 2 – n 2 + b 2 – 2bn + n 2

2 = a 2 + b 2 – 2bn.

Knowing that n = a * cos C, is replaced in the previous equation and the side value c is obtained:

2 = a 2 + b 2 – 2b * a * cos C.

According to the Law of the Cosines, its sides can be calculated as:

  • 2 = b 2 + c 2 – 2b * c * cos A.
  • 2 = a 2 + c 2 – 2a * c * cos B.
  • 2 = a 2 + b 2 – 2b * a * cos C.

There are cases where measurements of the sides of a triangle are unknown, but their height and angles are formed in the vertices. To determine the area in this case it is necessary to apply trigonometric ratios.

Knowing the angle of one node, the foot is identified and the appropriate trigonometric ratio is used:

For example, cathetus AB will be opposite to angle C, but next to angle A. Depending on the side or cathetus that corresponds to height, the other side is cleaned to get this value.

practice

First exercise

Calculate the area and height of the ABC scalene triangle, knowing that the sides are:

a = 8 cm.

b = 12 cm.

c = 16 cm.

The solution

As data is given measurements of three sides of the scalene triangle.

Because you don’t have a height value, you can determine the area by applying the Stork formula.

The first semiperimeter is calculated:

sp = (a + b + c) ÷ 2

sp = (8 cm + 12 cm + 16 cm) ÷ 2

sp = 36 cm ÷ 2

sp = 18 cm.

Now the values ​​in the Stork formula are replaced:

Knowing the area can be calculated relative height on the side b. From the general formula, clear what you have:

Area = (next to * h) ÷ 2

46, 47 cm 2 = (12 cm * h) ÷ 2

h = (2 * 46.47 cm 2 ) ÷ 12 cm

h = 92.94 cm 2 ÷ 12 cm

h = 7.75 cm.

Second exercise

Given a ABC scalene triangle, the size of which is:

  • Segment AB = 25 m.
  • Segment BC = 15 m.

At point B an angle of 50 ° is formed. Calculate the height relative to side c, the circumference and area of ​​the triangle.

The solution

In this case you have double-sided size. To determine the height, it is necessary to calculate the third side measurement.

Because the opposite angle from the given side is given, it is possible to apply the cosine law to determine the measurement of the AC side (b):

2 = a 2 + c 2 – 2a * c * cos B

Where:

a = BC = 15 m.

c = AB = 25 m.

b = AC.

B = 50 o .

Data replaced:

2 = (15) 2 + (25) 2 – 2 * (15) * (25) * cos 50

2 = (225) + (625) – (750) * 0.6427

2 = (225) + (625) – (482,025)

2 = 367,985

b = √367,985

b = 19.18 m.

Since you already have a three-sided value, calculate the perimeter of the triangle:

P = side a + side b + side c

P = 15 m + 25 m + 19, 18 m

P = 59.18 m

Now it is possible to determine the area by applying the Heron formula, but the first semiperimeter must be calculated:

sp = P ÷ 2

sp = 59.18 m ÷ 2

sp = 29.59 m.

Side and semiperimeter measurements are replaced in the Heron formula:

Finally, knowing the area, the relative height on the c side can be calculated. From the general formula, to clean it you must:

Area = (next to * h) ÷ 2

143.63 m 2 = (25 m * h) ÷ 2

h = (2 * 143.63 m 2 ) ÷ 25 m

h = 287.3 m 2 ÷ 25 m

h = 11.5 m.

Third exercise

In the ABC scalene triangle side b is 40 cm, side c is 22 cm, and at node A, angle 90 is formed o . Calculate the area of ​​the triangle.

The solution

In this case the measurement of the two sides of the ABC scalene triangle is given, as well as the angle formed at vertex A.

To determine the area it is not necessary to calculate the size of the side a, because through the trigonometric ratio the angle is used to find it.

Because the angle opposite to the height is known, this will be determined by the product on one side and the sine of the angle.

Replacing the area formula that you must:

  • Area = (next to * h) ÷ 2
  • h = c * cent A

Area = (b * c * cent A) ÷ 2

Area = (40 cm * 22 cm * 90 cents) ÷ 2

Area = (40 cm * 22 cm * 1) ÷ 2

Area = 880 cm 2 ÷ 2

Area = 440 cm 2 .

Reference

  1. Vlvaro Rendón, AR (2004). Technical Drawing: activity notebook.
  2. Angel Ruiz, HB (2006). CR Geometry Technology,.
  3. Angel, AR (2007). Pearson’s Basic Educational Algebra.
  4. Baldor, A. (1941). Havana Algebra: Culture.
  5. Barbosa, JL (2006). Euclidean Flat Geometry. Rio de Janeiro ,.
  6. Coxeter, H. (1971). Fundamentals of Mexican Geometry: Limusa-Wiley.
  7. Daniel C. Alexander, GM (2014). Basic Geometry for Students. Learning Cengage.
  8. Harpe, P. d. (2000). Topics in Geometric Group Theory. University of Chicago Press.

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