**I variant**

1. Construct a section of tetrahedron DABC by a plane passing through points A, B and F if F Є DC.

2. In the tetrahedron DABC: M is the midpoint of DC, K is the midpoint of AC, N is the midpoint of BC.

a) Construct a section of the tetrahedron by a plane passing through points M, K and N.

b) Find the perimeter of the section if DB=8cm, AD=6cm, AB=4cm.

c) Prove the parallelism of planes ADB and KMN.

3. Construct the section of parallelepiped ABCD1B1C1D1 by the plane passing through E, P and M if EA : ED = 1:5, PD : PD1 = 1 : 3, MC : MC1 = 4 :1.

4. All the faces of the parallelepiped ABCD1B1C1D1 are rectangles.

a) Construct a section of the parallelepiped by a plane passing through points D, M, P and C, if M is the midpoint of A1D1, P is the midpoint of B1C1.

b) Find the perimeter of the section if AB=3cm, AD=6cm, DD1=4cm.

c) Prove the parallelism of lines MD and PC.

5. Construct a section of tetrahedron DABC by a plane passing through E, K, and M if ЕЄАD, КЄDС, МЄАB.

**Control work on the theme: “The tetrahedron and the parallelepiped”.**

**variant II**

1. Construct a section of tetrahedron DABC by a plane, passing through points B, C, and K, if K Є AD.

2. In the tetrahedron DABC: M is the midpoint of AB, K is the midpoint of AC, N is the midpoint of AD.

a) Construct a section of the tetrahedron by a plane passing through points M, K, and N.

b) Find the perimeter of the section if DB=10cm, CD=8cm, BC=6cm.

c) Prove the parallelism of planes BCD and KMN.

3. Construct the section of parallelepiped ABCD1B1C1D1 by the plane passing through E, P, and F, if EЄ B1C1, PЄ A1B1, FA : FA1 = 1 : 4.

4. All sides of the parallelepiped ABCDD1B1C1D1 are rectangles.

a) Construct a section of the parallelepiped by a plane passing through points D, E, F and C, if E is the midpoint of AA1, F is the midpoint of BB1.

b) Find the perimeter of the section if AA1=12cm, DC=7cm, AD=8cm.

c) Prove the parallelism of lines ED and FC.

5. Construct a section of tetrahedron DABC by a plane passing through points K, M, and P if KЄAD, MЄBD, PЄBC.

**Control work on the theme: “The tetrahedron and the parallelepiped”.**

**I variant**

1. Construct a section of tetrahedron DABC by a plane, passing through points A, B, and F, if F Є DC.

2. In the tetrahedron DABC: M is the midpoint of DC, K is the midpoint of AC, N is the midpoint of BC.

a) Construct a section of the tetrahedron by a plane passing through points M, K and N.

b) Find the perimeter of the section if DB=8cm, AD=6cm, AB=4cm.

c) Prove the parallelism of planes ADB and KMN.

3. Construct the section of parallelepiped ABCD1B1C1D1 by the plane passing through E, P and M if EA : ED = 1:5, PD : PD1 = 1 : 3, MC : MC1 = 4 :1.

4. All the faces of the parallelepiped ABCD1B1C1D1 are rectangles.

a) Construct a section of the parallelepiped by a plane passing through points D, M, P and C, if M is the midpoint of A1D1, P is the midpoint of B1C1.

b) Find the perimeter of the section if AB=3cm, AD=6cm, DD1=4cm.

c) Prove the parallelism of lines MD and PC.

5. Construct a section of tetrahedron DABC by a plane passing through E, K, and M if ЕЄАD, КЄDС, МЄАB.

**Control work on the theme: “The tetrahedron and the parallelepiped”.**

**variant II**

1. Construct a section of tetrahedron DABC by a plane, passing through points B, C, and K, if K Є AD.

2. In the tetrahedron DABC: M is the midpoint of AB, K is the midpoint of AC, N is the midpoint of AD.

a) Construct a section of the tetrahedron by a plane passing through points M, K, and N.

b) Find the perimeter of the section if DB=10cm, CD=8cm, BC=6cm.

c) Prove the parallelism of planes BCD and KMN.

3. Construct the section of parallelepiped ABCD1B1C1D1 by the plane passing through E, P, and F, if EЄ B1C1, PЄ A1B1, FA : FA1 = 1 : 4.

4. All sides of the parallelepiped ABCDD1B1C1D1 are rectangles.

a) Construct a section of the parallelepiped by a plane passing through points D, E, F and C, if E is the midpoint of AA1, F is the midpoint of BB1.

b) Find the perimeter of the section if AA1=12cm, DC=7cm, AD=8cm.

c) Prove the parallelism of lines ED and FC.

5. Construct a section of tetrahedron DABC by a plane passing through points K, M, and P if KЄAD, MЄBD, PЄBC.

**Control work on the theme: “The tetrahedron and the parallelepiped”.**

**I variant**

1. Construct a section of tetrahedron DABC by a plane, passing through points A, B, and F, if F Є DC.

2. In the tetrahedron DABC: M is the midpoint of DC, K is the midpoint of AC, N is the midpoint of BC.

a) Construct a section of the tetrahedron by a plane passing through points M, K and N.

b) Find the perimeter of the section if DB=8cm, AD=6cm, AB=4cm.

c) Prove the parallelism of planes ADB and KMN.

3. Construct the section of parallelepiped ABCD1B1C1D1 by the plane passing through E, P and M if EA : ED = 1:5, PD : PD1 = 1 : 3, MC : MC1 = 4 :1.

4. All the faces of the parallelepiped ABCD1B1C1D1 are rectangles.

a) Construct a section of the parallelepiped by a plane passing through points D, M, P and C, if M is the midpoint of A1D1, P is the midpoint of B1C1.

b) Find the perimeter of the section if AB=3cm, AD=6cm, DD1=4cm.

c) Prove the parallelism of lines MD and PC.

5. Construct a section of tetrahedron DABC by a plane passing through E, K, and M if ЕЄАD, КЄDС, МЄАB.

**Control work on the theme: “The tetrahedron and the parallelepiped”.**

**variant II**

1. Construct a section of tetrahedron DABC by a plane, passing through points B, C, and K, if K Є AD.

2. In the tetrahedron DABC: M is the midpoint of AB, K is the midpoint of AC, N is the midpoint of AD.

a) Construct a section of the tetrahedron by a plane passing through points M, K, and N.

b) Find the perimeter of the section if DB=10cm, CD=8cm, BC=6cm.

c) Prove the parallelism of planes BCD and KMN.

3. Construct the section of parallelepiped ABCD1B1C1D1 by the plane passing through E, P, and F, if EЄ B1C1, PЄ A1B1, FA : FA1 = 1 : 4.

4. All sides of the parallelepiped ABCDD1B1C1D1 are rectangles.

a) Construct a section of the parallelepiped by a plane passing through points D, E, F and C, if E is the midpoint of AA1, F is the midpoint of BB1.

b) Find the perimeter of the section if AA1=12cm, DC=7cm, AD=8cm.

c) Prove the parallelism of lines ED and FC.

5. Construct a section of tetrahedron DABC by a plane passing through points K, M and P if KЄAD, MЄBD, PЄBC.