Lessons 45–46 · 2 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: parallelism
Textbook, Part 2: pp. 116–118
GoalApply knowledge of parallelism of lines and planes to problems on the tetrahedron, prism and pyramid.
Parallelism problems usually go in three steps. First, draw a picture and mark the given points and parallel lines. Second, find familiar properties of plane figures (triangle, trapezoid, parallelogram) such as a midline or similarity: for example, in triangle SAB the segment KL joining the midpoints of SA and SB is parallel to AB and half as long. Third, apply the criteria of parallelism: if KL ∥ AB and AB lies in plane ABC, then KL ∥ (ABC). A section of a pyramid by a plane parallel to the base is a polygon similar to the base: it lies in a plane parallel to it and its sides are parallel to the corresponding sides. The area of the section is proportional to the square of the similarity ratio.
Worked examples
SABC is a tetrahedron and K, L, M are the midpoints of SA, SB, SC. KL ∥ AB and LM ∥ BC (midlines). KL and LM intersect and are parallel to two lines of ABC, so KLM ∥ ABC. If AB = 14, KL = 7.
In the triangular prism ABCA₁B₁C₁, CC₁ ∥ AA₁, and AA₁ lies in face ABB₁A₁ while CC₁ does not. By the criterion CC₁ ∥ (ABB₁A₁): a lateral edge is parallel to the opposite lateral face.
Class activity
Show a section of a pyramid parallel to the base on a cardboard model (cutting with an adult’s help) or draw it, and check the similarity with the base by measuring the ratio of sides.
Practice
1
In triangle SAB, K and L are midpoints of SA and SB. How are KL and AB related?
KL ∥ AB and KL = AB/2.
2
In SABC, K and L are the midpoints of SA and SB and AB = 18. Find KL.
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3
A pyramid has base area 36. A plane parallel to the base cuts the height at 1/3 of its length from the apex. Find the area of the section.
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4
Why is KL ∥ (ABC)?
KL ∥ AB, AB lies in plane ABC and KL does not; this is the criterion for a line parallel to a plane.