See more detailed article here Method of finding cross-sectionThis article will explain the concept of section and two ways of determining the cross-section of a pyramid, that is, the root intersection method and the radial projection method.

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1. What is the cross-section of a shape

?Definition: The cross-section (or cross-section) of a figure H when cut by the plane (P) is the common part of the plane (P) and shape H. Find the cross-section ie find this cross-sectional shape, usually a polygon like triangle, quadrilateral…

2. What is the easiest way to determine the cross-section

?To determine the cross-section of a pyramid when cut by a plane, we have two main methods, the origin intersection method and the projection method. radial.

With problems related to cross-section, students need to master the following basic knowledge:

Concept of cross-section (section): Given shape T and plane (P), the plane part of (P) lying in T bounded by intersections generated by (P) intersecting some faces of T is called is the cross-section (section). Two distinct planes containing two parallel lines, respectively, their intersection, if any, is also parallel to those two lines or coincides with one of those two lines. Two distinct planes. are parallel to the same line, their intersection, if any, is also parallel to that line.

Ways to define planes: Know three noncollinear points; two intersecting straight lines; a point that lies outside a line; Two parallel lines.

Note.

Assume that the cross-sectional plane is (P), the polyhedron is T. Construction of the cross-section is a construction problem, but only the construction part and the argument part need to be stated if any. The vertex of the cross-section is the intersection of the plane (P) and the edges of the figure T, so the cross-section construction is essentially finding the intersection of (P) and the edges of T. The plane (P) may not intersect all the faces of T. The methods of section construction are given. depending on the hypothetical form of the title.

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Let’s practice with the following problem:

Exercise 1.

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Let S.ABC pyramid with M, N being the midpoints of SA, SB, respectively. P is the point on edge SC such that SP is larger than PC (i.e. MP is not parallel to AC). Determine the cross-section of the pyramid when cut by the plane (MNP).

The problems related to the cross-section are usually: Calculate the area of ​​the cross-section; find the position of the plane (P) so that the cross-section has the largest and smallest area; cross section divides the polyhedron into 2 parts with a given ratio. (or find the ratio between the 2 parts).

3. Some methods to find the fastest cross-section

Plane (P) for explicit form: Three noncollinear points, two intersecting lines, or a point outside a line…

Original intersection method.

First, find a way to determine the intersection of (P) with a face of T (this intersection is often called the origin). On this plane of T, find also the intersection of the origin and the sides of T. T aims to create some more common ground. Repeat this process with other faces of T until the cross-section is found.

Exercise 2.

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Given a pyramid S.ABCD whose base is a square (or parallelogram). Let M, N, P be the midpoints of BC, CD, SA respectively. Determine the cross-section of the pyramid when cut by the plane (MNP).