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Practice with lines and planes in space

Lessons 24 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
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Practice and applications: lines and planes in space

Textbook, Part 1: pp. 135–143
GoalApply what we know about lines and planes in space to the cube and the parallelepiped: decide mutual positions, construct a section and find its measurements, and explain practical situations.
New words
common point · umumiy nuqtaline of intersection · kesishish chizig‘iarea of a section · kesim yuzibeing coplanar · bir tekislikda yotish
Explanation

Before solving we draw a picture: hidden edges are dashed and the given points are marked. For a question on two lines we first ask whether they have a common point; if not, they are parallel when they lie in one face and skew otherwise. Two planes always intersect in a line, which we draw by finding two common points. To find the area of a section we first identify its shape: sections of a cube can be triangles, quadrilaterals and other polygons. Side lengths usually come from the Pythagorean theorem in right triangles: a face diagonal of a cube is a√2 and a space diagonal is a√3. These ideas work in practice too: a taut string gives a straight line, and a three-point support always rests on one plane.

Worked examples
A cube with edge 6 cm. The section through A, C and B₁ is an equilateral triangle with side equal to a face diagonal, 6√2 cm. Its area is (√3/4) · (6√2)² = (√3/4) · 72 = 18√3 cm².
A rectangular parallelepiped with edges 3, 4 and 12: the base diagonal is √(3² + 4²) = 5; the space diagonal is √(5² + 12²) = 13.
Class activity

Groups of three check with a taut string whether the four leg ends of a chair lie in one plane (stretch strings between opposite leg ends; if they meet, the points are coplanar). Do not use sharp or broken objects.

Practice
1
What is formed when two planes intersect?
2
How many other edges of a cube intersect a given edge?
3
A cube has edge 4 cm. Points M, N, L are the midpoints of AB, BC and BB₁; find the perimeter of triangle MNL.
4
A carpenter checks a board surface with a straight edge in several directions. Why is that enough?