MATHS

Planimetry (plane geometry)

Basic theoretical information

Triangle

When solving problems in geometry, in addition to all the geometric formulas and properties that will be given below, it is necessary to remember very well the basic formulas for trigonometry . Let’s start with a few basic properties of different types of angles:

  • Adjacent angles add up to 180 degrees.
  • Vertical angles are equal to each other.

We now turn to the properties of the triangle. Suppose there is an arbitrary triangle:

Arbitrary triangle

Then, the sum of the corners of the triangle :

Formula Sum of the angles of a triangle

Remember also that the sum of any two sides of a triangle is always greater than a third party . The area of ​​the triangle through two sides and the angle between them:

Formula The area of ​​a triangle through two sides and the angle between them

The area of ​​the triangle through the side and the height lowered onto it:

Formula The area of ​​a triangle through the side and the height lowered to it

The triangle semi-perimeter is as follows:

Formula triangle semi-perimeter

Heron’s formula for the area of ​​a triangle:

Heron formula for the area of ​​a triangle

The area of ​​the triangle through the radius of the circumcircle:

Formula Area of ​​a triangle through the radius of the circumscribed circle

Median formula (median – a line drawn through a certain vertex and middle of the opposite side in a triangle):

Median formula

Median properties:

  • All three medians intersect at one point.
  • The medians divide the triangle into six triangles of the same area.
  • At the intersection point, the medians are divided by a ratio of 2: 1, counting from the vertices.

The bisector property (a bisector is a line that divides a certain angle into two equal angles, that is, in half):

Formula Property of bisector

It is important to know:  The center of a circle inscribed in a triangle lies at the intersection of bisectors (all three bisectors intersect at this one point). Formula bisector:

Bisector formula

Bisector formula

The main property of the heights of a triangle (height in a triangle is a line passing through a certain vertex of the triangle perpendicular to the opposite side):

Formula Basic property of the heights of the triangle

All three heights in the triangle intersect at one point. The position of the intersection point is determined by the type of triangle:

  • If the triangle is acute, then the point of intersection of the heights is inside the triangle.
  • In a right triangle, heights intersect at the vertex of a right angle.
  • If the triangle is obtuse, then the point of intersection of the heights is outside of the triangle.

Formula height:

Formula height

Another useful property of the heights of the triangle:

Formula The property of the heights of a triangle

Cosine theorem :

Formula Cosines Theorem

Sinus Theorem :

Formula Sine Theorem

The center of the circle described near the triangle lies at the intersection of the middle perpendiculars. All three middle perpendiculars intersect at one this point. The middle perpendicular is the line drawn through the middle of the side of the triangle perpendicular to it.

The radius of a circle inscribed in a regular triangle:

Formula Radius of a circle inscribed in a regular triangle

The radius of a circle described around a regular triangle:

Formula Radius of a circle described around a regular triangle

Area of ​​a right triangle:

Formula Area of ​​a regular triangle

Pythagorean theorem for a right triangle ( c – hypotenuse, a and b – legs):

Formula Pythagorean theorem

The radius of a circle inscribed in a right triangle:

Formula Radius of a circle inscribed in a right triangle

The radius of a circle described around a right triangle:

Formula Radius of a circle described around a right triangle.

The area of ​​a right triangle ( h is the height lowered to the hypotenuse):

Formula Area of ​​a right triangle

Properties of the height lowered on the hypotenuse of a right triangle:

Formula Properties of height, lowered on the hypotenuse of a right triangle

Formula Properties of height, lowered on the hypotenuse of a right triangle

Formula Properties of height, lowered on the hypotenuse of a right triangle

Such triangles are triangles whose angles are respectively equal, and the sides of one are proportional to the similar sides of the other. In such triangles, the corresponding lines (heights, medians, bisectors, etc.) are proportional. Similar sides of similar triangles are opposite sides of equal angles. The coefficient of similarity is the number k , equal to the ratio of the similar sides of similar triangles. The ratio of the perimeters of such triangles is equal to the coefficient of similarity. The ratio of the lengths of bisectors, medians, heights and median perpendiculars is equal to the coefficient of similarity. The ratio of the areas of similar triangles is equal to the square of the similarity coefficient. Signs of similarity of triangles:

  • In two corners. If two corners of one triangle are respectively equal to two corners of another, then the triangles are similar.
  • On two sides and the corner between them. If two sides of one triangle are proportional to two sides of another, and the angles between these sides are equal, then the triangles are similar.
  • On three sides. If the three sides of one triangle are proportional to the three similar sides of the other, then the triangles are similar.

Trapezium

A trapezoid  is a quadrilateral in which exactly one pair of opposite sides is parallel. Trapezium centerline length:

Formula Length of trapezium midline

Trapezium area:

Formula Trapezium Area

Some trapezoid properties:

  • The middle line of the trapezoid is parallel to the bases.
  • The segment connecting the centers of the diagonals of the trapezium is equal to the half-difference of the bases.
  • In the trapezium of the midpoints, the intersection point of the diagonals and the intersection point of the side extensions are on the same line.
  • The diagonals of a trapezoid divide it into four triangles. The triangles whose sides are bases are similar, and the triangles whose sides are sides are equal in size.
  • If the sum of the angles at any base of the trapezium is 90 degrees, then the segment connecting the midpoints of the bases is equal to the half-difference of the bases.
  • In an isosceles trapezoid, the angles for any base are equal.
  • In an isosceles trapezoid, the diagonals are equal.
  • In an isosceles trapezoid, the height lowered from the apex onto a larger base divides it into two segments, one of which is equal to the half-sum of bases, and the other half-difference of the bases.

Parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel in pairs, that is, they lie on parallel lines. The area of ​​the parallelogram through the side and the height lowered onto it:

Formula The area of ​​the parallelogram through the side and the height lowered to it

The area of ​​the parallelogram through two sides and the angle between them:

Formula Parallelogram area across two sides and angle between them.

Some properties of the parallelogram:

  • The opposite sides of the parallelogram are equal.
  • The opposite angles of the parallelogram are equal.
  • The diagonal parallelograms intersect and the intersection point is divided in half.
  • The sum of the angles adjacent to one side is 180 degrees.
  • The sum of all the corners of the parallelogram is 360 degrees.
  • The sum of the squares of the parallelogram diagonals is equal to twice the sum of the squares of its sides.

Square

A square is a quadrilateral in which all sides are equal, and all angles are equal to 90 degrees. The area of ​​a square through the length of its side:

Formula Square Area through the length of its side

The area of ​​the square through the length of its diagonal:

Formula Square Area through the length of its diagonal

The properties of a square are all properties of a parallelogram, a rhombus and a rectangle at the same time.

Rhombus and Rectangle

A rhombus is a parallelogram, in which all sides are equal. Diamond square (the first formula is two diagonals, the second is through the length of the side and the angle between the sides):

Formula Diamond Area

Diamond properties:

  • The rhombus is a parallelogram. Its opposite sides are pairwise parallel.
  • Diagonal rhombus intersect at right angles and at the intersection point are divided in half.
  • Diagonal rhombus are the bisectors of its angles.

A rectangle is a parallelogram, in which all angles are right (equal to 90 degrees). The area of ​​the rectangle through two adjacent sides:

Formula Area of ​​a rectangle through two adjacent sides

Rectangle properties:

  • The diagonals of the rectangle are equal.
  • A rectangle is a parallelogram – its opposite sides are parallel.
  • The sides of the rectangle are at the same time its heights.
  • The square of the diagonal of a rectangle is equal to the sum of the squares of its two not opposite sides (according to the Pythagorean theorem).
  • A circle can be described near any rectangle, and the diagonal of the rectangle is equal to the diameter of the circumcircle.

Freehand shapes

The area of ​​an arbitrary convex quadrilateral in two diagonals and the angle between them:

Formula The area of ​​an arbitrary convex quadrilateral in two diagonals and the angle between them

Connection of the area of ​​an arbitrary figure, its half-perimeter and the radius of the inscribed circle (it is obvious that the formula is valid only for figures in which a circle can be entered, that is, including for any triangles ):

Formula Connection of the area of ​​an arbitrary figure, its half-perimeter and the radius of the inscribed circle

The generalized theorem of Thales: Parallel straight lines cut off on proportional segments.

The condition under which it is possible to inscribe a circle in a quadrilateral:

The condition under which it is possible to inscribe a circle in a quadrilateral

The condition under which it is possible to describe a circle around a quadrilateral:

The condition under which it is possible to describe a circle around a quadrilateral

The sum of the angles of the n -gon:

Formula Sum of n-gon angles

Central angle of a regular  n -gon:

Formula Central angle of a regular n-gon

The area of ​​a regular  n -gon:

Formula Area of ​​a regular n-square

Circle

Tangent property:

Tangent property

Chord property:

Chord property

The theorem on proportional segments of chords:

Formula Theorem on proportional segments of chords

Theorem on tangent and secant:

Formula Theorem on tangent and secant

The theorem of two secants:

Formula Theorem on two secants

The theorem on the central and inscribed angles (the size of the central angle is twice the size of the inscribed angle, if they are based on a common arc):

Formula Central and inscribed theorems theorem

Property of inscribed angles (all inscribed angles based on a common arc are equal to each other):

Corner property

Property of the central corners and chords:

Formula Property of Central Angles and Chords

Property of the central corners and secants:

Formula Property of Central Angles and Interceptors

Circumference :

Formula Circumference

Circumference:

Formula Circumference

Circle Area :

Formula Circle Area

Sector Area:

Formula Sector Area

Ring area:

Formula Square Ring

Circular segment area:

Formula Circular Segment Area

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