Solving problems according to ready-made drawings "regular triangular pyramid". Calculation of the areas of figures according to ready-made drawings

correct triangular pyramid Solving problems according to ready-made drawings MBOU Verkhnyakovskaya secondary school Mathematic teacher: Martynenko L.N. DABC-regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Task #1: Find DO

Hints:
Find D.K.
Apply Triangle Medians Property
Apply the Pythagorean theorem to find DO

DABC-regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Task number 2: Find P bases.

Hints:
Apply the law of cosines

DABC-regular pyramid, DO perpendicular (ABC) Problem №3: O1 and O2 are the points of intersection of medians of triangles ABD and BCD respectively O1O2=2. Find S bases.

Hints:
Consider triangles KDM and DO1O2
Find KM
Using the midline property of a triangle, find the side of the triangle

DABC-regular pyramid, DO ┴ (ABC),CK ┴ AB, AM ┴ BC , BN ┴ AC . Task #4: Find DO

Hints:
Use Triangle Median Property
Apply the Pythagorean theorem to find height

DABC-regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Task #5: Find the angle DKC

Hints:
Apply the Triangle Bisector Property

DABC-regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Task #6: Find DO

Hints:
What element needs to be found to calculate DO?
Use Triangle Median and Ratio Property in Right Triangle

DABC-regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Task number 7: Find the apothem DM.

Hints:
Apply Triangle Median Property to Find OM

DABC-regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC , BN ┴ AC. Task #8: Find COS

Hints:
Use Triangle Median and Ratio Property in Right Triangle

DABC-regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Task #9: Find COS

Hints:
Use the median property of a triangle and the ratio in a right triangle

DABC-regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Task #10: Find the SPDL

Hints:
Find DO

DABC-regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Task #11: Find SPQL

Hints:
Write the formula for the area of a triangle
Find PL from similarity triangles ABC and APL
Find QL from similar triangles ADC and AQL
Find the height of triangle PQL using the Pythagorean theorem

DABC-regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Task #12: Find the SDKC

Hints:
Write the formula for the area of a triangle
Find CK
Use the median property of a triangle to find CO
Find the height of triangle CDK

The manual offered to the reader's attention contains more than 1000 multi-level tasks and exercises on the main topics of the geometry (planimetry) program of grades 7-9, arranged in 3 sets according to ready-made drawings. Grade 7 contains 12 tables, grade 8 - 25, 9-12 tables.
These exercises enable the teacher to solve and repeat a much larger amount of material within a minimum of time, thereby increasing the pace of work in the classroom.
In addition, brief theoretical information on the geometry course of grades 7-9 is given, accompanied by definitions, theorems, basic properties and necessary reference materials. Solutions and instructions are given for the most difficult problems.
The manual is addressed to mathematics teachers, tutors, students - future teachers, students general education schools, lyceums, colleges, as well as graduates to prepare for the GIA and the Unified State Examination.

Four wonderful points of the triangle.
Each triangle has 4 points associated with it:
1) point of intersection of medians;
2) the point of intersection of the bisectors;
3) point of intersection of heights (or their extensions);
4) the point of intersection of the medial perpendiculars to the sides.
These four points are called the remarkable points of the triangle.

The height of a triangle is the length of the perpendicular dropped from any of its vertices to opposite side or its continuation.
AT obtuse triangle(Fig. 29) two heights fall on the extension of the sides and lie outside the triangle, and the third inside.
In an acute triangle (Fig. 30), all three heights lie inside the triangle.
In a right triangle, the legs simultaneously serve as heights (Fig. 31).

CONTENT
Preface 3
Section I. Brief theoretical information 5
Section II. Exercises in tables 28
VII class
Table 1. Adjacent corners 28
Table 2. Vertical angles 30
Table 3. Signs of equality of triangles 32
Table 4. Perimeter isosceles triangle 36
Table 5. Properties of an isosceles triangle 38
Table 6. Signs of parallel lines 40
Table 7. Properties of angles with parallel lines 45
Table 8. Angles of a triangle 47
Table 9. Angles of a triangle 48
Table 10. Some properties of right triangles 52
Table 11. Signs of equality of right triangles 56
Table 12. Distance from point to line 57
VIII class
Table 1. Definition and features of a parallelogram 59
Table 2. Properties of a parallelogram 61
Table 3. Properties of a parallelogram 64
Table 4. Parallelogram 66
Table 5. Parallelogram 68
Table 6. Trapeze 69
Table 7. Trapeze 72
Table 8. Area of a rectangle 73
Table 9. Area of a parallelogram 76
Table 10. Area of a triangle 79
Table 11. Area of a trapezoid 82
Table 12. Pythagorean Theorem 86
Table 13 Definition similar triangles 93
Table 14. Signs of similarity of triangles 98
Table 15. Signs of similarity of triangles 102
Table 16 middle line triangle 105
Table 17 Proportional segments in a right triangle 108
Table 18. Relations between sides and angles in a right triangle 110
Table 19. Relations between sides and angles in a right triangle 112
Table 20
Table 21. Central and inscribed angles 118
Table 22
Table 23
Table 24. Vectors 138
Table 25
IX class
Table 1. Vector coordinates 148
Table 2. The simplest problems in coordinates 149
Table 3. Application of the coordinate method to problem solving 152
Table 4. Equation of a circle 154
Table 5. Equation of a straight line 156
Table 6. Solution of triangles. Triangle area 158
Table 7. Solution of triangles. Sine theorem 162
Table 8. Solution of triangles. Cosine theorem 164
Table 9 Scalar product vectors 168
Table 10. Circumference. Arc length 171
Table 11. Circle area 176
Table 12
Section III. Solutions to some problems 181
VII class 181
VIII class 183
IX class 198
Answers 213.

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Download the book Geometry, Tasks on ready-made drawings for preparing for the GIA and the Unified State Examination, grades 7-9, Balayan E.N., 2013 - fileskachat.com, fast and free download.

Geometry, grade 7-9, Tasks on ready-made drawings for preparing for the GIA and the Unified State Examination, Balayan E.N., 2013
Tutor in geometry to prepare for the State Academic Examination and Unified State Examination, grades 7-11, Balayan E.N., 2012
Geometry, grades 7-9, workshop on planimetry, getting ready for the GIA, Glazkov Yu.A., Egupova M.V., 2014
Geometry trainer, Grade 7, to the textbook by Atanasyan L.S. etc. “Geometry. Grades 7-9”, Federal State Educational Standard, Glazkov Yu.A., Yegupova M.V., 2019

Solving problems on ready-made drawings on the topic: "

triangles

Geometry, 7th grade

To the textbook by L.S. Atanasyan

mathematics teacher of the highest category

MOU "Upshinsky basic comprehensive school"

Orsha district of the Republic of Mari El

Some properties

right triangles

Property 1 0 . Sum sharp corners right triangle is 90 0 .

Property 2 0 . The leg of a right triangle lying opposite an angle of 30 0 equal to half of the hypotenuse.

Property 3 0 . If the leg of a right triangle is half the hypotenuse, then the angle opposite that leg is 30 0 .

Property 4 0 . In a right triangle, the median drawn from the vertex right angle equal to half of the hypotenuse.

Property 5 0 . If the median of a triangle is equal to half of the side to which it is drawn, then this triangle is right-angled.