# Math 5: Geometry and Trigonometry

*A Biomedical science student, certificated in english, spanish, french and japanese language, who also writes diverse topic articles.*

Content• Geometric figures

• Perimeters, areas and volume

• Congruence of triangles

• Similar triangles

• Right triangles and Pythagorean theorem

• Law of Sines

• Law of cosines

• Analytic geometry

• Conical sections

• Three-dimensional geometry

## Geometric Figures

The Geometric figures are anyfigure that is formed in space that meets certain characteristics. For its study many factors are considered, the number of sides it has, the length of those sides, its internal angles, and that it is closed (that is, that all its sides are connected by a vertex).

## Perimeters, areas and volume

The perimeter is known as the measure of the contour of any figure, which means, the sum of all of its sides together. The area is the space formed inside of the figure in a two-dimensional plane, for example if we have a square of 4 cm per side, the amount of cm2 in its interior would be equal to 16. Finally, the volume is the space within a figure but in the three-dimensional plane, that occurs when you have a composed figure, such as a pyramid, a cylinder, a sphere, a prism or any figure with a height. For example, if you have a prism which's base is the square of the previous example, but with a height of 10 cm; the amount of cm3 in its interior would be equivalent to 160. However, there are several ways to know the perimeter, area or volume of a figure beyond counting the units corresponding to a figure, these forms are functions in which the values of perimeter area or volume are established according to the length of its sides, or the angles in its interior.

In general, to calculate the perimeter of most geometric figures, multiplications and/or sums are used, however the values that are multiplied are different depending on the figure, in the case of figures such as squares, rectangles or triangles it is easy use sums since their sides are visible and easily measurable, however for circles obtaining the perimeter is more complicated, since it requires other values and the formulas P = πd (where "d" is the diameter of the circle).

To calculate the area of any figure a formula is necessary, since these (for the most part) require measurements beyond its sides. The formulas to calculate these data are in the following table.

Figure | Formula | Specifications |
---|---|---|

Square | A = (s)(s) | Where s is the value of the side of the square |

Rectangle | A = (L) (W) | Where L is the lenght and W is the width of the rectangle |

Triangle | A = 1/2(b)(h) | Where b is the base of the triangle and h is the height |

Diamon | A = (b)(h) | Where b is the length of the base and h is the height |

Trapezoid | A= h[(b1 + b2)/2] | Where b1 and b2 are the lengths of the parallel sides and h the distance (height) between the parallels. |

Circle | A = πr2 | Where r is the radius |

To obtain the volume of any figure, its base must be considered, that is, the geometric figure from which its formed, in the case of a cylinder it would be a circle, from a prism the shape of its base, square, rectangular, hexagonal, etc. In this way the volume of 3 different three-dimensional figures can be obtained. Prisms, pyramids and spheres.

Figure | Formula | Specifications |
---|---|---|

Prisms | V = (A)(h) | Where A is the Area of the base figure and h is the height of the prism |

Pyramids | V = 1/3(A)(h) | Where A is the Area of the base figure and h is the height of the prism |

Spheres | V = 4/3 (πr3) | Where r is the radius |

## Congruence of triangles

When lines intersect, especially when a transversal intersects a pair of parallel lines, the intersections form angles with special relationships between them. These angular relationships are used to explain how to construct parallel or perpendicular lines, which in turn help us to bisect angles and line segments. In addition, parallel sides are also used to reveal more properties of parallelograms.

For example, although all of the lines in the following figure appear to be parallel, only two of them are parallel.

Transversal lines help us identify parallel lines in two ways.

- Lines are parallel only if they have congruent alternate interior angles.
- Lines are parallel only if they have congruent corresponding angles.

In this case there are two congruent angles labeled, but they are not alternate interior angles or corresponding angles. Therefore, the missing angle measurements must be filled in to obtain more information and be able to determine it.

Two figures are determined to be congruent only if one can be mapped with reference to the other exactly, preserving distances and angle measures, so that all sides and angles are congruent. That means that one way to decide if a pair of triangles are congruent is to measure all the sides and angles and them to be equal.

Although with only 3 measurements it can be shown that two triangles are congruent. To demonstrate congruence with another class of figures, we can divide the figure, as long as it is a polygon, into triangles and show that the triangles inside are congruent.

## Triangle congruence criteria

Criteria | Specifications |
---|---|

Side-Side-Side (SSS) | When the three corresponding pairs of sides are congruent, the triangles are congruent. |

Side-Angle-Side (SAS) | When two pairs of corresponding sides and the angles between them are congruent, the triangles are congruent. |

Angle-side-angle (ASA) | When two pairs of corresponding angles and the sides between them are congruent, the triangles are congruent. |

Angle-Angle-Side (AAS) | When two pairs of corresponding angles and one pair of corresponding sides (not between the angles) are congruent, the triangles are congruent. |

Hypotenuse - cathetus (HC) | When the hypotenuses and a pair of corresponding sides of right triangles (cathetus) are congruent, the triangles are congruent. |

## Similar triangles

Similar triangles, as their name indicates, are those whose sides and angles, although not congruent, are similar or proportional, that is, a triangle with a side of 5 cm, one with 4 and one with 3 is similar to one with the same shape but whose sides measure 2.5, 2 and 1.5 correspondingly.

## Right triangles and Pythagorean theorem

A right triangle is one that has an angle of 90 °, these triangles differ from the others because they have specific ways of being solved, which are more related to trigonometry than to geometry.

To begin with, the **Pythagorean theorem** is the formula determined by a^{2} + b^{2} = c^{2}, where a and b are the lengths of the cathetus of a right triangle, and c is the length of the hypotenuse. The theorem means that if we know at least two of the lengths of any right triangle, we can determine the length of the other side using the formula.

The terms used to designate the sides of a right triangle must be defined, the **hypotenuse** is the longest side and it is always the one that is opposite the right angle (the 90 ° angle).

The other two sides are called the **opposite and adjacent cathetus**. The names are given by their relationship to an angle. The opposite leg is the side that is opposite the given angle, while the adjacent leg is the side that is next to the given angle, and is not the hypotenuse.

To obtain a missing lateral length in a right triangle when we only know one length and the measure of an angle (other than the right angle), a series of relation between sides and angles is used, these relations are established as formulas with which also Angle measures can be obtained from the lengths of two sides.

These relationships between the sides of a right triangle are called trigonometric ratios, the three common trigonometric ratios (and most commonly used in right triangles) are sin, cos, and tan.

## Law of Sines

There are two more ways to solve right (or not right) triangles when you have only certain data, for example, when you have 2 sides, and the angle opposite one of them; when you have the measures of 2 angles of a triangle, and the side opposite one of them; or when 2 angles of the triangle are known and a side that is not opposite to any of them. In these cases the theorem or law of sines is used.

On the other way, in cases where we have other data, such as when we have the measure of an angle and the sides adjacent to it; and when you have the measure of the 3 sides of a triangle but no angle. In these cases the theorem or law of cosines can be used.

## Analytic geometry

Analytical geometry uses algebraic methods and equations for the study of geometric problems, it studies figures, their distances, their areas, points of intersection and angles of inclination. The basic idea of this discipline is the establishment of a correspondence between the points of a plan, this can be done in many ways, the points that are far from O by units (one, two, three ...) are indicated on the axis and they are assign a number. In this way, the positive values (+ a) are attributed to the units located to the right of O in the x axis and the negative values (-a) to the symmetric segment of the same with respect to the point O. Thus, we establish a correspondence between the points of the x-axis and the real numbers.

If a line parallel to the x-axis is drawn from the y-axis that is corresponding to b and another line that is parallel to the y-axis through the point corresponding to a on the x-axis, its point of intersection (P) is designated as P (a, b ). Therefore, given a pair of real numbers a and b, there is only one possible point, the one that has a as the abscissa and b as the ordinate. Likewise, if we choose a point in the plane we can draw from it a single point parallel to each coordinate axis. These lines will cut the axes at the points marked a and b, numbers that will constitute the pair corresponding to point P. We will then say that the coordinates of P are (a, b).

The two axes divide the plane into four quadrants. These are called the first, second, third and fourth quadrant, the first being the one in the upper right part to point 0, the second the one in the upper left part, the third, the lower left, and, the fourth, the lower right. The points of the first quadrant have positive coordinates, those of the second quadrant are negative on the abscissa (x) and positive on the ordinate (y); those of the third are negative, and those of the fourth are positive on the axis x and negative on the y axis.

In mathematics, analytical geometry plays an important role in calculus. It is a fundamental tool for finding tangents, points, lengths, areas, and volumes.

## Conical sections

Conic section is how all of the curves resulting from the different intersections between a cone and a plane are called. They are classified into four types: ellipse, parabola, hyperbola, and circumference.

The generatrix is any one of the oblique lines.

The vertex is the central point where the generatrices intersect.

The leaves are the two parts where the vertex divides the conical surface of revolution.

Conic section is called the intersection curve of a cone with a plane that does not pass through its vertex, different conic sections can be obtained.

**Ellipse:** It is the section produced in a conical surface of revolution by a plane oblique to the axis, which is not parallel to the generatrix and which forms with it an angle greater than that formed by the axis and the generatrix.

α <β <90º

The ellipse is a closed curve.

**Circumference:** The circumference is the section produced by a plane perpendicular to the axis.

β = 90º

The circumference is a particular case of ellipse that completes 360 °.

**Parabola:** is the section produced in a conical surface of revolution by a plane oblique to the axis, being parallel to the generatrix.

α = β

The parabola is an open curve that continues to infinity.

**Hyperbola:** is the section produced in a conical surface of revolution by a plane oblique to the axis, forming with it an angle smaller than that formed by the axis and the generatrix, therefore it affects the two sheets of the conical surface.

α> β

The hyperbola is an open curve that continues indefinitely and consists of two separate branches.

## Three-dimensional geometry

Three-dimensional geometry is one that is responsible for volume calculations in 3D figures. That is, pyramids, prisms, spheres, cylinders, among others. whose formulas are shown in the area, perimeter and volume calculations section.

**© 2021 Daniela Alejandra Rodríguez Cerda**