cadaVR anatomy by Mozaik Education

Properties of notable planar objects

Triangles

The usual notations concerning triangles can be seen in Figure 69. We will use this terminology in the following theorems.

I. Theorems concerning the angles and sides of triangles

THEOREM: The sum of the interior angles of a triangle is \latex{ 180 }º, that is

\latex{\alpha +\beta +\gamma =180°.}

THEOREM: Any exterior angle of a triangle is equal to the sum of the two opposite interior angles, that is,

\latex{\alpha '=\beta +\gamma ;\beta '=\gamma +\alpha ;\gamma '=\alpha +\beta .}

Triangle inequality: the sides of any triangle satisfy that the sum of the length of any two sides is always greater than the length of the third side, that is,

\latex{a+b\gt c;}

\latex{b+c\gt a;}

\latex{c+a\gt b,}

or, in other words,

\latex{a\gt |b-c|;}

\latex{b\gt |c-a|;}

\latex{c\gt |a-b|.}

THEOREM: Two sides of a triangle are equal in length if and only if the angles opposite to said sides are equal. In short,

\latex{a=b\Leftrightarrow \alpha =\beta .}

THEOREM: In any triangle, the angle opposite to a longer side is bigger and vice versa, the side opposite to a larger angle is longer. In short,

\latex{a\gt b\Leftrightarrow \alpha \gt \beta .}

LAW OF SINES: The ratio of the sides of any triangle is equal to the ratio of the sines of the opposite angles, that is,

\latex{a:b:c=\sin \alpha :\sin \beta :\sin \gamma .}
The same ratios written in pairs:
\latex{\frac{a}{b}=\frac{\sin \alpha }{\sin \beta }, \frac{b}{c}=\frac{\sin \beta }{\sin \gamma },\frac{c}{a}=\frac{\sin \gamma }{\sin \alpha }.}

LAW OF COSINES: Every triangle satisfies the following:

\latex{a^{2}=b^{2}+c^{2}-2\times b\times c\times \cos \alpha ,}
\latex{b^{2}=c^{2}+a^{2}-2\times c\times a\times \cos \beta ,}
\latex{c^{2}=a^{2}+b^{2}-2\times a\times b\times \cos \gamma .}

Let \latex{ a } and \latex{ b } denote the length of the legs and \latex{ c } be the length of the hypotenuse of a right angled triangle (Figure 70), then by applying the law of cosines for \latex{ c } we have

\latex{c^{2}=a^{2}+b^{2}-2\times a\times b\times \cos 90°=a^{2}+b^{2}.}

The Pythagorean theorem is a special case of the law of cosines.

PYTHAGOREAN THEOREM: In any right angled triangle the sum of the squares of length of the legs equals the length of the hypotenuse squared, that is,

\latex{c^{2}=a^{2}+b^{2}.}

Since \latex{0°\lt \gamma \lt 180°} and in this interval \latex{\cos \gamma =0} if and only if \latex{\gamma =90°,} if a triangle satisfies \latex{c^{2}+a^{2}+b^{2}} then the angle \latex{\gamma} opposite to the side \latex{ c } is a right angle. Therefore the reverse of the Pythagorean theorem is also true.

THEOREM: If the sum of two sides' length squared equals the length of the third side squared in a triangle, then that triangle is right angled.

The Pythagorean theorem and its reverse can be stated as one:

THEOREM: A triangle is right angled if and only if the sum of the squares of two sides' lengths equals the length of the third side squared.

*It is a consequence of the law of cosines that for an obtuse triangle \latex{(\gamma \gt 90°)} \latex{c^{2}\gt a^{2}+b^{2},} while for an acute triangle \latex{(\gamma \lt 90°)} \latex{c^{2}\lt a^{2}+b^{2}.}

II. Notable lines and points in triangles

THE INTERNAL ANGLE BISECTORS OF A TRIANGLE

THEOREM: The internal angle bisectors of a triangle intersect in a single point. This point is the centre of the triangle's inscribed circle (incentre). (Figure 71)

ANGLE BISECTOR THEOREM: Any internal angle bisector of a triangle divides the opposite side such that the ratio of the two segments created equals to the ratio of the other two sides of the triangle. (Figure 72)

\latex{ A }

\latex{ B }

\latex{ C }

\latex{ M }

\latex{ r }

Figure 71

\latex{\alpha}

\latex{\frac{\alpha }{2}}

\latex{\frac{CS}{SB}=\frac{b}{c}}

\latex{ A }

\latex{ B }

\latex{ C }

\latex{ S }

\latex{ b }

\latex{ c }

Figure 72

THE PERPENDICULAR BISECTORS OF THE TRIANGLE'S

THEOREM: The perpendicular bisectors of a triangle intersect in one single point. This point is the centre of the circumscribed circle (circumcentre). (Figure 73)

If the triangle is acute, then the circumcentre is inside the triangle; the circumcentre of a right angled triangle is the midpoint of the hypotenuse (Thales' theorem), and the circumcentre of an obtuse triangle is outside of the triangle. (Figure 74)

\latex{ O }

Figure 74

ALTITUDES OF A TRIANGLE

DEFINITION: The altitude segment of a triangle is a line segment starting at a vertex and ending on the line of the opposite side such that it is perpendicular to the opposite side (Figure 75). The line incident to the altitude segment is called the altitude.

acute triangle

right angled triangle

obtuse triangle

\latex{ A }

\latex{ m_{a} }

\latex{ a }

\latex{ B }

\latex{ m_{b} }

\latex{ a }

\latex{ b }

\latex{ A }

\latex{ m_{a} }

\latex{ a }

Figure 75

THEOREM: The altitudes of a triangle intersect in a single point called the orthocentre of the triangle. (Figure 76)

\latex{ A }

\latex{ B }

\latex{ C }

\latex{ M }

\latex{ A }

\latex{ B }

\latex{ C=M }

\latex{ A }

\latex{ B }

\latex{ C }

\latex{ M }

Figure 76

MEDIANS OF THE TRIANGLE

DEFINITION: The median of a triangle is the segment between the midpoint of one side and the opposite vertex.

THEOREM: For every triangle, any two medians of the triangle intersect such that the point of intersection divides both medians into two parts with a ratio of \latex{ 1:2 } where the larger part contains the vertex as endpoint. (Figure 77)

The following is a straight consequence of the previous theorem:

THEOREM: The medians of any triangle intersect in a single point called the centroid of the triangle. For all medians the centroid is the trisecting point farther from the vertex.

MIDLINES OF THE TRIANGLE

DEFINITION: The segment joining two midpoints of a triangle is called the midline of the triangle. (Figure 78)

THEOREM: The segment joining two midpoints of a triangle (the midline) is parallel to the third side and its length is half the length of the third side. (Figure 79)

III. Special triangles

THE ISOSCELES TRIANGLE

DEFINITION: A triangle is called isosceles when it has two equal sides (legs).

It is easy to prove that for any triangle, any one of the following three properties is equivalent to the property in the definition. (Figure 80)

It is symmetric across a line.
It has two angles of equal size.
The perpendicular bisector of one side (base) is incident to the third vertex.

THE EQUILATERAL (REGULAR) TRIANGLE

DEFINITION: A triangle is called regular if its sides are of equal length.

It follows from the definition using a previous theorem that the angles of the regular triangle are also equal, all being \latex{ 60 }º.

The regular triangle has three axes of symmetry. (Figure 81)

The length of the altitude segment of the regular triangle with sides of length \latex{ a } is \latex{\frac{a\times \sqrt{3} }{2},} its area is \latex{\frac{a^{2}\times \sqrt{3} }{4}}. (Figure 82)

THE RIGHT ANGLED TRIANGLE

DEFINITION: A triangle is called right angled if it has an angle of size \latex{ 90 }º. (Figure 83)

Thales' theorem and the Pythagorean theorem, along with their reverses, give us two necessary and sufficient conditions for a triangle being right angled.

THALES' THEOREM AND ITS REVERSE: A triangle is right angled if and only if the centre of its circumscribed circle is the midpoint of one side.

PYTHAGOREAN THEOREM AND ITS REVERSE: A triangle is right angled if and only if the sum of the squares of the lengths of two sides equals the length of the third side squared.

\latex{ a }

\latex{ b }

\latex{ c }

\latex{ 90 }º

Figure 83

\latex{ A }

\latex{ B }

\latex{ C }

\latex{ T }

\latex{ a }

\latex{ b }

\latex{ c }

\latex{ m }

\latex{ q }

\latex{ p }

Figure 84

Using the similarities of the right angled triangles \latex{ ATC, CTB } and \latex{ ACB } one can state the following two well-known theorems concerning proportions in right angled triangles.

GEOMETRIC MEAN THEOREM (ALTITUDE THEOREM): In any right angled triangle, the length of the altitude to the hypotenuse is the geometric mean of the two segments determined by the foot of the altitude on the hypotenuse, that is,

\latex{m=\sqrt{p\times q} .}

THEOREM: In any right angled triangle, the length of a leg is the geometric mean of the hypotenuse and the segment determined by the foot of the altitude on the hypotenuse incident to the leg, that is,

\latex{a=\sqrt{c\times p}} and \latex{b=\sqrt{c\times q}.}

IV. Determining the area of a triangle based on different data

Let \latex{ a, b, c } denote the lengths of the sides, \latex{\alpha ,\beta ,\gamma} the corresponding interior angles and \latex{m_{a},m_{b},m_{c}} the corresponding altitudes of a triangle.

Let us denote the half of the circumference \latex{\left(s=\frac{a+b+c}{2} \right)} by \latex{ s }, the radius of the incircle by \latex{ r }, the radius of the circumscribed circle by \latex{ R }. The area \latex{ A } of the triangle can be computed using the following formulae:

\latex{A=\frac{a\times m_{a} }{2}=\frac{b\times m_{b} }{2}=\frac{c\times m_{c} }{2};}

\latex{A=\frac{a\times b\times \sin \gamma }{2}=\frac{b\times c\times \sin \alpha }{2}=\frac{c\times a\times \sin \beta }{2};}

\latex{A=s\times r.}

\latex{*}

\latex{A=\frac{a\times b\times c}{4\times R};}

\latex{A=\frac{a^{2}\times \sin \beta \times \sin \gamma }{2\times \sin \alpha };}

\latex{A=2\times R^{2}\times \sin \alpha \times \sin \beta \times \sin \gamma ;}

\latex{A=\frac{R^{2} }{2}\times (\sin 2\alpha +\sin 2\beta +\sin 2\gamma ).}

Quadrilaterals

DEFINITION: A quadrilateral is said to be convex, if all of its angles are convex, and concave, if it has a concave (reflex) angle. (Figure 85)

Figure 85

THEOREM: The sum of the sizes of the interior angles is \latex{ 360 }º for any quadrilateral.

I. Special quadrilaterals (Figure 86)

trapezoid

parallelogram

rhombus

convex

concave

rectangle

square

kite

Figure 86

A trapezoid is a quadrilateral which has a pair of parallel sides.

A parallelogram is a quadrilateral whose opposite sides are parallel.

*It can be verified that the following statements concerning parallelograms are equivalent, therefore any of them alone suffices to define parallelograms.

A parallelogram is a centrally symmetric quadrilateral.
The opposite sides of a parallelogram are equal in length.
The opposite sides of a parallelogram are parallel.
The opposite angles of a parallelogram are equal in size.
The sum of any two neighbouring angles in a parallelogram is \latex{ 180 }º.
The diagonals of a parallelogram halve each other.
In a parallelogram, one pair of opposite sides are equal in length and parallel.

A rhombus is a quadrilateral with sides of equal length.

A kite is a quadrilateral whose sides can be grouped into two pairs of adjacent sides with equal length. (Kites are quadrilaterals with a symmetry across a diagonal.)

A rectangle is a quadrilateral whose interior angles are all right angles.

A square is a quadrilateral whose sides are of equal length and angles are of equal size.

The diagrams in Figure 87 illustrate the relation between the classes of quadrilaterals defined above.

trapezoids

parallelograms

rhombuses

kites

rhombuses

parallelograms

rectangles

rhombuses

squares

Figure 87

II. Midlines in quadrilaterals

DEFINITION: The segment connecting the midpoints of two opposite sides of a quadrilateral is called the midline of the quadrilateral. (Figure \latex{ 88 })

THEOREM: The midlines of any quadrilateral cut each other in half.

The midlines of parallelograms and trapezoids have special properties.

THEOREM: Any midline of a parallelogram is parallel and equal in length to two sides of the parallelogram. (Figure 89)

Using reflection through a point and the previous theorem concerning the midline of a parallelogram, the following theorem can be shown:

THEOREM: The midline connecting the midpoints of the legs in a trapezoid is parallel to the bases of the trapezoid, and its length is the arithmetic mean of the length of the two bases. (Figure 90)

\latex{F_{b}F_{d}=\frac{a+b}{2}}

\latex{F_{b}F_{d}||AB||CD}

\latex{ A }

\latex{ B }

\latex{ D' }

\latex{ A' }

\latex{ C }

\latex{ D }

\latex{ a }

\latex{ c }

\latex{ d }

\latex{ a }

\latex{ c }

\latex{ b }

\latex{ F_{d} }

\latex{ F_{b} }

\latex{ F_{d} }

Figure 90

III. Tangential quadrilaterals, cyclic quadrilaterals

*DEFINITION: A quadrilateral is said to be tangential, if it has an inscribed circle.
(Figure 91)

Using the equality of the lengths of the tangent line segments from a point to a circle, the following theorem can be shown:

*THEOREM: A convex quadrilateral is tangential if and only if the sums of the lengths of the opposite sides are equal.

It follows from the theorem, that, for example, all kites are tangential quadrilaterals.

*DEFINITION: A quadrilateral is said to be cyclic, if it has a circumscribed circle.
(Figure 92)

The following theorem can be proven using the theorem concerning central angles and inscribed angles:

*THEOREM: A quadrilateral is cyclic if and only if the sum of its opposite angles is \latex{ 180 }º. Trapezoids which are symmetric across a line (isosceles trapezoids) are cyclic.

Polygons

DEFINITION: A polygon is convex, if all of its angles are convex, and concave, if it has a concave (reflex) angle. (Figure 93)

*Convexity and concavity are properties that can be defined for arbitrary planar of \latex{ 3 }-dimensional object.

*DEFINITION: An object is said to be convex, if the segment connecting any two of its points is also contained in the object. (Figure 94)

*DEFINITION: An object is said to be concave, if it is not convex, that is, there exists a segment connecting two points of the object which is not entirely contained in the object. (Figure 95)

Since there are \latex{n – 3} diagonals starting at an arbirtaty point of a convex \latex{ n }-sided polygon, the following theorem holds:

THEOREM: The number of diagonals in a convex \latex{ n }-sided polygon is

\latex{\frac{n\times (n-3)}{2}.}

The following theorem is straightforward to prove for convex polygons, and it is true for any polygon.

THEOREM: The sum of the sizes of angles in an \latex{ n }-sided polygon is

\latex{(n-2)\times 180.}

*An inmediate consequence of the above theorem is that the sum of the sizes of the external angles for any convex polygon is \latex{ 360 }º.

DEFINITION: A polygon is called regular if all of its sides are of equal length and all of its angles are of equal size.

THEOREM: The size of an interior angle of a regular polygon with \latex{ n } sides is

\latex{\frac{(n-2)\times 180°}{n}.}

The circle

We have already summarized some of the knowledge about the circle as a notable point set. In this section we will refresh some more results.

I. Tangents of a circle

There are two tangent lines to any circle from any point outside the circle.

THEOREM: The tangent segments to a circle from a point outside the circle are equal in length. (Figure 96)

II. Central angle, arc, sector; arc-length of angles

DEFINITION: If the vertex of an angle is the centre of a circle, then the angle is a central angle of the circle. (Figure 97)

THEOREM: If \latex{\alpha} and \latex{\beta} are two central angles in a circle with the lengths of the corresponding arcs being \latex{i_{\alpha }} and \latex{i_{\beta }}, then \latex{\frac{i_{\alpha } }{i_{\beta } }=\frac{\alpha }{\beta },} that is, the length of the arc is proportional to the size of the central angle.

THEOREM: If \latex{\alpha} and \latex{\beta} are two central angles in a circle with the lengths of the corresponding sectors being \latex{t_{\alpha }} and \latex{t_{\beta }}, then \latex{\frac{t_{\alpha } }{t_{\beta } },} that is, the area of the sector is proportional to the size of the central angle.

The definition of the arc-length of angles:

DEFINITION: The size of the central angle belonging to an arc of length \latex{ r } in a circle with radius \latex{ r } is \latex{ 1 } radian.

Corollaries:

Using that the circumference of the circle with radius \latex{ r } is \latex{2r\pi}, the size of the full angle is \latex{2\pi}.
The arc-length of an angle of size \latex{\alpha} a measured in radian in a circle with radius \latex{ r } is:\latex{i_{\alpha }=r\times \alpha} (rad).
Using that the area of the circle with radius \latex{ r } is \latex{r^{2}\pi ,} the area of the sector belonging to a central angle of size \latex{\alpha} measured in radian in a circle with radius \latex{ r } is:\latex{t_{\alpha }=\frac{\alpha \times r^{2} }{2}=\frac{i_{\alpha \times r} }{2}.}

The formulae for the conversion of the size of an angle between degrees and radians are the following:

\latex{\alpha °=\frac{360°}{2\pi (rad)}\times \alpha (rad)=\frac{180°}{\pi (rad)}\times \alpha (rad),}

\latex{\alpha (rad)=\frac{2\pi (rad)}{360°}\times \alpha °=\frac{\pi (rad)}{180°}\times \alpha °.}

*III. Central and inscribed angles, visual angle

*DEFINITION: If the vertex of an angle is a point of a given circle and its arms are incident to either two chords of the circle or a chord and a tangent of the circle, then the angle is called an inscribed angle of the circle. (Figure 98)

\latex{\alpha}

\latex{\beta}

\latex{ O }

\latex{ A }

\latex{ B }

\latex{ C }

\latex{ D }

Figure 98

*THEOREM: In a given circle the size of an inscribed angle is half of the size of the central angle belonging to the same arc. (Theorem of central and inscribed angles.)

More generally,

*THEOREM: In circles with the same radius, the ratio of the size of an central and a inscribed angle belonging to arcs of the same length is \latex{ 2 : 1 }.

Thales' theorem is a special case of the theorem of central and inscribed angles: in this case, the size of the central angle is \latex{ 180 }º while that of the inscribed angle is \latex{ 90 }º.
(Figure 99)

There is exactly one central and infinitely many inscribed angles to a given arc in a given circle, thus the next theorem follows easily from the one above:

*THEOREM: The inscribed angles belonging to a given arc of a given circle are equal in size. (Theorem of inscribed angles.) (Figure 100)

More generally,

*THEOREM: The inscribed angles belonging to arcs of equal length in circles with the same radii are equal in size.

*DEFINITION: From a point \latex{ P }, the segment \latex{ AB } is visible under angle \latex{\alpha} if \latex{APB\measuredangle =\alpha }. (Figure 101)

The theorem of inscribed angles using the concept of viewing angle:

*THEOREM: The given chord \latex{ AB } of a given circle can be seen at equal angles from the (internal) points of the arc \latex{ AB }. (Figure 102)

The following theorem is a consequence of the theorem of inscribed angles.

*THEOREM: The set of points in the plane from which a given segment \latex{ AB } of the plane is visible under a given angle \latex{\alpha} \latex{(0°\lt \alpha\lt 180°)} is two arcs positioned in a symmetric fashion on the two sides of the line \latex{ AB }. (Theorem of visual angles.) (Figure 103)

\latex{0\lt \alpha \lt 90°}

\latex{\alpha =90°}

\latex{90°\lt \alpha \lt 180°}

\latex{ O_{1} }

\latex{ O_{2} }

\latex{ O }

\latex{ O_{1} }

\latex{ O_{2} }

\latex{ A }

\latex{ B }

\latex{ A }

\latex{ B }

\latex{ A }

\latex{ B }

Figure 103

Exercises

{{exercise_number}}. Following the common notation, the lengths of the sides and the sizes of angles in the triangle \latex{ ABC } are, respectively, \latex{ a, b, c } and \latex{\alpha ,\beta ,\gamma .} Determine the lengths of the remaining sides and the sizes of the remaining angles if

\latex{a=5\,cm,\beta =75°,\gamma =65°;}

\latex{b=7.5\,cm,c =5.2\,cm,\beta =95°;}

\latex{a=11\,cm, b=3.5\,cm,\gamma =45°;}

\latex{a=8.1\,cm, b=9.3\,cm, c=6\,cm.}

{{exercise_number}}. Construct a triangle from the length of one side and the lengths of the other two altitudes \latex{(a,m_{b},m_{c} ).} What are the criteria of constructibility?

{{exercise_number}}. The length of the hypotenuse in a right angled triangle is \latex{ 20\,cm }. The altitude belonging to the hypotenuse divides it into parts with ratio \latex{ 1 : 5 }. Find the lengths of the legs, the sizes of the acute angles and the area of the triangle.

{{exercise_number}}. A triangular field was offered for purchase to a sport club. The lengths of the sides of the triangle are \latex{AB=238.9\,m;BC=249\,m;AC=151.8\,m.}

Find the size of the angle \latex{\alpha} at vertex \latex{ A } and the area of the field.

A rectangular sports pitch will be built on the field such that its area is maximal and one of its sides is incident to the side \latex{ AB }. Determine the lengths of the sides of said rectangle and find its area.

{{exercise_number}}. The vertices of a triangle divide its circumscribed circle with a radius of \latex{ 2\,cm } into three arcs with a ratio of \latex{ 5:6:7 }. Find

the sizes of the interior angles;

the lengths of the sides and the area of the triangle;

the areas of the segments of the circle determined by the triangle.

{{exercise_number}}. In a quadrilateral, the internal bisectors of the angles determine a smaller quadrilateral inside the original one. Compute the sizes of the angles in the smaller quadrilateral if the sizes of three internal angles in the original quadrilateral are \latex{ 105 }º, \latex{ 80 }º, \latex{ 125 }º. What can be said about the quadrilateral determined by the bisectors? Generalize the observation.

{{exercise_number}}. What can be said about the quadrilateral whose midlines are

equal in length;

perpendicular?

{{exercise_number}}. Determine the number of sides of the polygon with sum of the sizes of the interior angles being

less than \latex{ 1,300 }º but more than \latex{ 1,100 }º;

less that \latex{ 3,000 }º but more than \latex{ 2,000 }º.

{{exercise_number}}. How many sides does a polygon has if it has \latex{ k } times as many diagonals as sides (where \latex{ k } is a positive integer)?

{{exercise_number}}. The size of the interior angles of a \latex{ 10 } sided polygon measured in degrees are consequtive elements of an arithmetic progression with difference being \latex{ 6 }º. Find the sizes of the smallest and largest angle in the polygon.

{{exercise_number}}. A trapezoid is both cyclic and tangential quadrilateral. Prove that the radius of the inscribed circle is the geometric mean of its two bases.

Mathematics 12.

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