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Open lesson: “Dihedral angles” for students in grades 10-11 studying geometry using the textbook by L.S. Atanasyan
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Instructions for working with the presentation:
Slides are displayed using the mouse. You can start working from any slide. You can select part of the slides. You can copy the necessary material.
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Dihedral angles. 10th grade 2008
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Lesson objectives:1. Expand the concept: “Angle” 2. Derive the definition of dihedral angles. 3. Learn to measure dihedral angles4. Learn to apply the properties of dihedral angles when solving problems.
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Repetition.1. Definition of linear angle.2.Three perpendicular theorem.3.Slopes and projection.4.Definition of trigonometric functions.4. Properties of a right triangle.
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We display the angles gradually, at the command of the mouse, so we repeat the definition and properties Linear angle (acute, right, obtuse) Vertical angles Adjacent angles Central angle Inscribed angle.
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Perpendicular, oblique and projection. Theorem of three perpendiculars. Properties of obliques and projections. Repeat these questions in the problems.
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B S A K N Perpendicular, oblique and projection are related by the Pythagorean theorem. The theorem of three perpendiculars for the straight line KS. Plane ABC KS Equal inclined ones have …….. Large inclined………
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A B C D V H P N A B C D E F M H S O P R Find the angle between straight line HD (AO) and the plane of the base and the side face
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A D C B F Draw a perpendicular to DC and AD from point F ABCD – square, rhombus. How are perpendicular, oblique and oblique projection related to each other?
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A B C D F Where can you see the three perpendicular theorem?
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Task.
A perpendicular BM is drawn through vertex B of square ABCD. It is known that MA=4cm MD=5cm, Find the distance from M to the plane; Distance between MV and DC. A B C D M
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The main part of the lesson.
Practical tasks: Everyone took a file sheet, bent it into two unequal parts, and concluded that two intersecting half-planes with a common straight line are called a dihedral angle. How to measure it? Let's draw a common straight line, remember the axiom of planes, Let's mark a point on the edge. Let's draw perpendiculars to the edge from a given point in each face. We bend along the edge again and conclude that the angles are different, which means they need to be distinguished, how? We take scissors and make a cut along the perpendiculars, insert the sheet into the crack and see the linear angle. We look through the slides that provide answers to the proposals received. Let us define the measurement of dihedral angles. We show double angles on models of pyramids, prisms and on tables.
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Dihedral angles It is known that the measure of a dihedral angle is the measure of its linear angle. If we mark a point on the edge of a dihedral angle on each face and draw rays from this point perpendicular to the edge, we obtain a linear angle. M
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The point on the edge can be arbitrary...
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Definition:
α β B A C M N P
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It is sometimes convenient to construct a linear angle of a dihedral angle as follows: from some point A, we drop face α onto edge a AC┴a, the perpendicular to the other face AB┴β CB will be the projection of AC onto the plane β. Since AC┴a, then BC┴a by the inverse theorem about 3 perpendiculars. ACB is the linear angle of a dihedral angle with edge a. A B C a α β
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Perpendicular planes. Two intersecting planes are called perpendicular if the angle between them is 90°.
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Properties:
If a plane passes through a line perpendicular to another plane, then such planes are perpendicular.
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Problem solving:
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Notes on problem solving.
You can solve on computers using “Autofigures” You can solve on an “interboard”. Can be projected directly onto a regular board or whiteboard. We display the conditions of the problem on the screen and complete the drawing and solve it directly on the frame. Each student can save the solution to the problem, and the teacher will then evaluate it. You can display student solutions on a common screen and consider different methods.
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Point M is located on one of the faces of a dihedral angle equal to 30. The distance from the point to the edge of the dihedral angle is 18 cm. Calculate the distance from the projection of point M on the second face to the edge of the dihedral angle.
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The segments AC and BC lying on the faces of a right dihedral angle are perpendicular to its edge. Calculate the distance between points A and B if AC=10cm, BC=24cm.
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Point K, on the face of a dihedral angle, is removed from the other face by 12 cm, and from the edge by Calculate the value of the dihedral angle.
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Point A is located on the edge of a dihedral angle equal to each other. On its faces there are perpendiculars to the edges AB and AC, equal to 10 cm and 8 cm, respectively. Calculate the distance between points B and C.
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Find the distance from point D to line AB, if AC = CB = 10, AB = 16, CD = 6. Draw a perpendicular from point D to line AB. Find the dihedral angle at edge AB. ▲ABC, CD╨ABC D
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▲ABC, CD ╨ ABC). Find the distance from point D to straight line AB, (find the value of the dihedral angle at edge AB) straight ACB, AC = 15, CB = 20, CD = 35. A D
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Points M and K lie on different faces of a right dihedral angle. The distance from these points to the edge is 20 cm and 21 cm. Calculate the distance between the segments of the MC and the edge of the dihedral angle.
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The ends of the segment lie on the faces of the dihedral angle and are 6 cm away from its edge. The distance between this segment and the edge is 3 cm. Calculate the dihedral angle.
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Point K is 8 cm away from each side of the equilateral triangle ABC, AB = 24 cm. Calculate the value of the dihedral angle whose edge is the straight line BC, and whose faces contain points K and A.
K A V S A V S
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a) Plane M passes through side AD of square ABCD. Diagonal BD forms an angle of 45 degrees with plane M. Find the angle between the plane of the square and the plane M. b) The plane M passes through side AD of the square ABCD and forms an angle of 30 degrees with the plane. Find the angle that diagonal BD makes with plane M.
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The base of the pyramid PABCD is a rectangle ABCD, the sides of which are equal. The planes RAB and RBC are perpendicular to the plane ABC, and the plane PAC is inclined to it at an angle. Find the height and volume of the pyramid.
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Property of a trihedral angle.
If two plane angles are equal, then their common edge is projected onto the bisector of the third plane angle. A B C D
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All faces of the parallelepiped are equal rhombuses, with side a and an acute angle. Find the height of the parallelepiped.
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Answer:
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*The base of the pyramid is a rhombus. Two side faces are perpendicular to the plane of the base and the dihedral angle formed by them is 120°; the other two faces are inclined to the base plane at an angle of 30°. The height of the pyramid is h. Find the total surface area of the pyramid.
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MABCD - given pyramid, ABCD - rhombus; (ABM)┴(ABC) and (MSV)┴(ABC), means MV┴ABC). MB=H,ABC - linear angle of the dihedral angle with edge MB, ABC=120°. A B C D
The circles are equal. Find the area of the parallelogram. Part. Diagonal. Quadrangle. Parallelogram. Angles. Centers of circles. Circle. Proof. Triangles. Two circles. Property of a parallelogram. Height of the parallelogram. Geometry. Square. Area of a parallelogram. Properties of a parallelogram. Equality of segments. Dots. Tasks. Tangent to a circle. Acute angle. Middle line. Signs of a parallelogram.
“Dihedral angle, perpendicularity of planes” - All six faces are rectangles. The distance between intersecting lines. A sign of perpendicularity of two planes. Find the distance. Linear dihedral angle. Find the angle. A plane perpendicular to a line. Planimetry. Dihedral angles. Line a is perpendicular to the plane. Edge of a cube. Parallelepiped. Section. Planes ABC1 and A1B1D are perpendicular. Find the tangent of the angle. Diagonal.
“Corollaries from the axioms of stereometry” - Section of geometry. The intersection of a line and a plane. Flat and straight. Planes. Construct an image of a cube. How many faces pass through one, two, three, four points. Explanation of new material. Draw a straight line. Proof. Solution. Oral work. Statements. Axioms of stereometry and some consequences from them. What is stereometry? Axioms of planimetry. Find the line of intersection of the planes.
“The concept of a pyramid” - Faces of a pyramid. Test questions. Lateral ribs of the pyramid. Wonders of Giza. Polyhedron. Equal angles. Pyramid in economics. Travel route. At the base of the pyramid is the mastaba. Side edge. Egyptian pyramids. Pyramids in chemistry. The base of the pyramid. Step pyramids. Model of a modern industrial enterprise. A virtual journey into the world of pyramids. Side rib. The structure of the methane molecule. Adjacent side faces.
“Examples of central symmetry” - Patterns on carpets. Segment. An angle with a given degree measure. Plane. A segment of a given length. Central symmetry in a six-pointed star. Central symmetry. Central symmetry in squares. Hotel "Pribaltiyskaya". Chamomile. Examples of symmetry in plants. Straight. Central symmetry in a rectangular coordinate system. Central symmetry in transport. Axioms of stereometry. Central symmetry in zoology.
“Axioms of stereometry, grade 10” - Axioms of stereometry. A, B, C? one straight line A, B, C? ? ? - the only plane. A plane passes through two intersecting lines, and only one. Problem Given a tetrahedron MABC, each edge of which is 6 cm. Name the line along which the planes intersect: A) (MAB) and (MFC) B) (MCF) and (ABC). Corollaries from the axioms of stereometry. 4. Calculate the lengths of segments AK and AB1, if AD=a. 2. Find the length of segment CF and the area of triangle ABC.
5. Circle image:
The image of a circle with center at point O1 is an ellipse with center at point O, belonging to the projection plane α
The common perpendicular of two intersecting linescalled a segment with ends on these lines, perpendicular to each of them.
Distance between crossing linesis called the length of their common perpendicular. It is equal to the distance between parallel planes passing through these lines.
Angle between intersecting linesThe angle between intersecting lines parallel to the given intersecting lines is called.
Generalized three perpendicular theorem
Any straight line on a plane perpendicular to the projection of an inclined one onto this plane is also perpendicular to the inclined one.
And vice versa: if a straight line in a plane is perpendicular to an inclined one, then it is also perpendicular to the projection of the inclined one.
The angle between a straight line and a plane called the angle between a straight line and its projection on a plane (angle φ).
The angle between two intersecting planescalled the angle between the straight line of intersection of these planes with
a plane perpendicular to the line of intersection of these planes (angle φ‘).
Area of orthogonal projection of a polygon onto a planeis equal to the product of its area and the cosine of the angle between the plane of the polygon and the projection area.
Problem 1. Through the point O of the intersection of the diagonals of the square ABCD, a perpendicular MO of length 15 cm is drawn to its plane. Find the distance from point M to the sides of the square if its side is 16 cm.
Answer: 17 cm.
Problem 2. A segment AS equal to 12 cm is perpendicular to the plane of triangle ABC, in which AB=AC=20 cm, BC=24 cm. Find the distance from point S to straight line BC.
Answer: 20 cm.
Problem 3. To the plane of a rectangle ABCD, the area of which is 180 cm2, a perpendicular SD is drawn, SD = 12 cm, BC = 20 cm. Find the distance from point S to the sides of the rectangle.
Answer: 12 cm, 12 cm, 15 cm, 4 34 cm.
Problem 4. Leg AC of a right triangle is equal to a, angle B is equal to φ. Through the vertex of the right angle, a perpendicular MC of length a is drawn to the plane of this triangle. Find the distance from the ends of the perpendicular to the hypotenuse.
Answer: a cosϕ; a 1+ cos2 ϕ .
Problem 5. In triangle ABC, sides AB = 13 cm, BC = 14 cm, AC = 15 cm. From vertex A, a perpendicular AD of length 5 cm is drawn to its plane. Find the distance from point D to side BC.
Answer: 13 cm.
Problem 6. A perpendicular MC of length 7 cm is drawn to the plane of a rhombus ABCD, in which Ð A = 45°, AB = 8 cm. Find the distance from point M to the sides of the rhombus.
Answer: 7 cm, 7 cm, 9 cm, 9 cm.
Task 7. Construct common perpendiculars to straight lines AB and CD on the image of the cube.
Problem 8. A plane α is drawn through side AC of equilateral triangle ABC. The angle between the height BD of the triangle and this plane is equal to φ. Find the angle between line AB and plane α.
Answer: arcsinç |
sinϕ ÷ . |
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Problem 9. Through the center O of a regular triangle ABC is drawn to its plane
perpendicular to MO. AB=a 3. The angle between straight line MA and the plane of the triangle is 45°. Find the angle between the planes: 1) AMO and VMO; 2) IUD and ABC.
Answer: 1) 60°; 2) arctg 2.
Problem 10. The planes of equilateral triangles ABC and ABD are perpendicular. Find the angle:
1) between straight line DC and plane ABC; between planes ADC and BDC.
Answer: 1) 45°; 2) arccos 1 5 .
Problem 11. Prove the theorem on the projection area of a polygon for the case when the polygon is a triangle in which none of the sides are parallel to the projection plane.
Problem 12. The edge of the cube is equal to a. Find the cross-sectional area of the cube by a plane passing through the top of the base at an angle of 30° to this base and intersecting all the side edges.
Answer: 2 3 a 2 .
Problem 13. The sides of the rectangle are 20 and 25 cm. Its projection onto the plane is similar to it. Find the perimeter of the projection.
Answer: 72 cm or 90 cm.
Problem 14. An isosceles triangle with a height of 16 cm is bent along the midline MN, parallel to the base AC, so that vertex B is 4 cm away from the plane of the quadrilateral ACNM.
a) Find the angle between planes AMC and MBN;
b) Construct the linear angle of the dihedral angle BMNC and find the angular measure if the orthogonal projection of vertex B onto the plane of the quadrilateral AMNC lies outside its boundaries;
c) Compare the angular measures of the dihedral angle BMNC and the angle BMA; d) Find the distance from point B to straight line AC;
e) Find the distance from straight line MN to plane ABC;
f) Construct the intersection line of the AMB and BNC planes.
3. Self-control tasks
1. The edge of a cube is 10 cm. Find the distance between lines a and b.
2. Through vertex A of triangle ABC, a straight line a is drawn, perpendicular to the plane of the triangle. Find the distance between lines a and BC if AB = 13 cm, BC = 14 cm, AC = 15 cm.
Answer: 12 cm.
3. A perpendicular KD is drawn to the plane of the square ABCD. The side of the square is 5 cm. Find the distance between the lines: 1) AB and KD; 2) KD and AC.
Answer: 1) 5 cm; 2) 5 2 2 cm.
4. The angle between planes α and β is 30°. Point A, lying in the α plane, is 12 cm away from the line of intersection of the planes. Find the distance from point A to the β plane.
Answer: 6 cm.
5. Through the center O of the square ABCD a perpendicular SO is drawn to its plane. The angle between the straight line SC and the plane of the square is 60°, AB = 18 cm. Find the angle between the planes ABC and BSC.
Answer: arctg 6.
6. A square with side 4 2 cm is bent along a straight line that passes through the midpoints of M and N sides DC and BC, so that vertex C is removed from the plane
AMN by 1 cm.
a) find the angle between the planes ADM and CMN;
b) construct the linear angle of the dihedral angle BMNC and find its angular measure if the orthogonal projection of vertex C onto the plane of the pentagon ABNMD lies beyond its boundaries;
c) compare the angular measures of the dihedral angle BMNC and the angle CNB; d) find the distance from point C to straight line BD;
e) find the distance from straight line MN to plane BDC;
f) construct the line of intersection of the BNC and DMC planes.
Answer: a) 30°; d) 2 × 2 + 3 cm; d) 2 - 3 cm.
7. Vertices A and D of the parallelogram ABCD lie in the α plane, and the other two lie outside this plane, AB = 15 cm, BC = 19 cm. The projections of the diagonals of the parallelogram onto the α plane are 20 cm and 22 cm. Find the distance from side BC to plane α.
Directions: Use the theorem on the sum of the squares of the diagonals of a parallelogram.
Answer: 12 cm.
8. Point M is removed from each side of an isosceles trapezoid at a distance of 12 cm. The bases of the trapezoid are 18 cm and 32 cm. Find the distance from point M to the plane of the trapezoid.
Answer: point M lies in the plane of the trapezoid.
9. Through vertex A of rectangle ABCD, an inclined AM is drawn to the plane of the rectangle, making an angle of 50° with sides AD and AB. Find the angle between this inclined plane and the plane of the rectangle.
Answer: 32°57’.
10. The ends of the segment AB=25 cm lie on the faces of a dihedral angle equal to 60°. From points A and B, perpendiculars AC and BD are dropped onto the edge of the dihedral angle, AC = 5 cm, BD = 8 cm. Find CD.
Answer: 24 cm.
Lesson No. 7
Lesson topic: “Cartesian coordinate system in space”
- consolidate students' school knowledge about the rectangular coordinate system in space;
- systematize knowledge about the equations of figures in space;
- consolidate skills in solving problems on drawing up equations of geometric images in space.
1. Brief summary of theoretical material
t.O – origin of coordinates; Ox – abscissa axis; Оу – ordinate axis; Оz – applicate axis. xy , xz u yz – coordinate planes
Distance between two points
Coordinates of the midpoint of the segment
The figure F is given by this equation in rectangular coordinates, if a point belongs to the figure F if and only if the coordinates of this point satisfy the given equation. This means that 2 conditions are met:
1) if the point belongs to the figure F, then its coordinates satisfy the equation;
2) if the numbers x, y, z satisfy this equation, then the point with such coordinates belongs to the figure F.
Equation of a sphere A sphere is a set of points in space that are distant from a given point by
specified positive distance. In this case, this point is called the center of the sphere, and this distance is its radius.
A sphere of radius R with center at point A (a;b;c) is given by the equation (by definition)
(x - a) 2 + (y - b) 2 + (z - c) 2 = R 2.
If the center of the sphere coincides with the origin of coordinates, then a=b=c=0 and the equation of the sphere has the form: x 2 + y 2 + z 2 = R 2.
Plane equation
Theorem. A plane in space is specified in a system of rectangular coordinates x, y, z by an equation of the form Ax+By+Cz+D=0, provided that A2 +B2 +C2 >0.
The converse statement is also true: the equation Ax+By+Cz+D=0, provided that A2 +B2 +C2 >0 defines a plane in space in the rectangular coordinate system.
Equation of a line
A straight line in space is the line of intersection of two planes.
Ð A1 x + B1 y + C1 z + D1 = 0; í î A2 x + B2 y + C2 z + D2 = 0.
If the line AB passing through points A (x1 ;y1 ;z1 ) and B (x2 ;y2 ;z2 ) is not parallel to any coordinate plane, then its equation has the form:
x−x1 |
y−y1 |
z − z1 |
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2. System of tasks for classroom training
Problem 1. The side of a cube is 10. Find the coordinates of its vertices.
Problem 2. Find the perimeter of triangle ABC if A(7;1;-5), B(4;-3;-4), C(1;3;-2).
Answer: 14 + 26.
Problem 3. Do three points A, B, C lie on the same line if A(3;2;2), B(1;1;1),
Answer: Yes.
Problem 4. Which of the points – A(2;1;5) or B(-2;1;6) – lie closer to the origin? Answer: Point A.
Problem 5. Given points K(0;2;1), P(2;0;3) and T(-1;y;0). Find a value of y such that the condition is satisfied: CT = RT.
Answer: -3.
Problem 6. Find the coordinates of the midpoints of the sides of triangle ABC, if A(2;0;2),
B(2;2;0), C(2;2;2).
Answer: A1 (2;2;1), B1 (2;1;2), C1 (2;1;1).
Problem 7. Find the length of the median AM of triangle ABC if A(2;1;3), B(2;1;5),
Answer: AM=1.
Problem 8. Which of the following equations are equations of a sphere:
a) x 2 − y 2 |
x 2 + y 2 + z 2 =1; |
c) x 2 + y 2 + z 2 = a 2; |
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d) x 2 + y 2 |
1+ x ; |
2x 2 + y 2 + z 2 =1; |
e) x 2 + y 2 + z 2 − 2x + 3y − 4z =1? |
Problem 9. Write the equations of the plane passing through: a) the Ox axis and the point A(1;1;1);
b) points O(0;0;0); A(1;2;-3) and B(2;-2;5).
Problem 10. The plane and sphere are given by the equations 4x+3y–4=0 and x2 +y2 +z2 –2x+8y+8=0. Does the center of the sphere belong to this plane?
Problem 11. Write an equation for a straight line passing through points A(1;3;2) and
Find their intersection points.
Problem 13. Find the distance from vertex D of the tetrahedron ABCD to its face ABC,
if AC=CB=10, AB=12, DA=7, DB= 145, DC= 29.
Answer: 3.
Problem 14. Find the length of edge AD of tetrahedron ABCD, if AB=AC=BC=10,
DB=2 29, DC= 46 and the distance from the vertex D to the plane of the face ABC is equal to
Answer: 214 or 206.
3. Self-control tasks
1. Given points K(0;1;1); P(2;-1;3) and T(-1;y;0). Find a value of y such that the condition is satisfied: CT = RT.
2. Given points A (1;2;3) and B (3;-6;7). Find the coordinates of the midpoint of segment AB.
3. Find the coordinates of a point that lies on the Oy axis and is equidistant from points A (4;-1;3) and B (1;3;0).
4. Find the points equidistant from the points A(0;0;1), B(0;1;0), C(1;0;0) and at a distance of 2 from the yz plane.
5. Points A(a;0;0), B(0;a;0), |
С(0;0;а) – vertices of the triangle. Find the coordinates |
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the intersection points of the medians of this triangle. |
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Belongs |
sphere whose equation |
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x 2 + y 2 + z 2 − 2x + 4y − 6z − 2 = 0? |
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Find a point |
sphere intersections, |
given |
equation x 2 + y 2 + z 2 − 4 x = 12 s |
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8. Write an equation for a plane parallel to the xy plane and passing through point A(2;3;4).
9. Points O(0;0;0); A(3;0;0); B(0;4;0) and O 1 (0;0;5) – vertices of a rectangular parallelepiped. Write down equations for the planes of all its faces.
10. Write down equations for a straight line passing through points A(1;1;2) and B(-3;2;7).
11. At what distance from the base of the cube is a segment of length b located parallel to the base, if one end of the segment lies on the diagonal of the cube, the other on the diagonal of the side face that intersects it? Length of cube edge a.
Answer: (2a ± 5b 2 − a 2) ÷ 5.
12. ABCDA1 B1 C1 D1 – rectangular parallelepiped, AB=BC=a, AA1=2a. Find the length of the segment MK, parallel to the face ABB1 A1, if M AD1, K DB1, AM:AD1 = 2:3.
Answer: a 3 5 .
Lesson No. 8
Topic of the lesson: “Vectors in space and the vector method for solving stereometric problems”
- generalize and deepen students’ school knowledge about vectors and actions on them;
- continue studying the vector method for solving planimetric and stereometric problems; a for " a, b.
Property 2: (xa) × b = x(a × b) for " a, b, x. Property 3: (a + b) × c = a × c + b × c for " a, b, c.
Two special cases:
1) a = b; a × a = a2 = a 2 .
2) a × b = 0 if and only if the vectors a and b are perpendicular. If a or b is a zero vector, then by definition it is perpendicular to any vector.
If a =(a1;a2;a3); b =(b1 ;b2 ;b3 ), then a × b = a 1 × b 1 + a 2 × b 2 + a 3 × b 3 .
