Geometry in space: corrected 2nd grade math exercises.

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1 Series of exercises on geometry in space

The exercises on geometry in space in the second year of secondary school (2de) provide you with in-depth and detailed knowledge. Indeed, you will learn to solve problems in the simplest way. In addition, this chapter requires the use of the geometric materials necessary to properly handle the proposed exercises.

Series of exercises on geometry in space

Exercise 1:

Let ABCD be a tetrahedron and I, J two points belonging respectively to the edges [AB] and [BC] such that (IJ) is not parallel to (AC). Let P be the plane passing through B and parallel to the plane (IJD). The goal of the exercise is to draw the intersection of the plane P with the plane (ACD).

1) The line (IJ) intersects the line (AC) at K. Draw the line of intersection of the planes (ACD) and (IJD). Justify.

2) Let D be the line of intersection of the plane P and the plane (ABC). Why is D parallel to (IJ)? Draw D.

3) The line D intersects the line (AC) at L. Let D’ be the line of intersection of the plane P and the plane (ACD).
Why is D’ parallel to (DK)? Draw D’.

Exercise 2:

Let be a pyramid of vertex S whose base is a quadrilateral ABCD.
We place I on [SA] such that $\vec{SI}=\frac{1}{3}\vec{SA}$ , and J on [SD] such that $\vec{SJ}=\frac{1}{3}\vec{SD}$

1) Draw $\Delta$ the intersection of the (CIJ) plane and the base plane. Justify this construction.

2) Determine without justification the section of the pyramid through the plane (CIJ)

Exercise 3:

Let be a pyramid SABCD such that (AB) and (CD) intersect at E.

1) Determine the intersection of the planes (SAB) and (SDC)

2) A plane P parallel to (ES) cuts (SA) at I, (SB) at J, (SC) at K, (SD) at L.
Show that (IJ) and (KL) are parallel.

Exercise 4 Geometry in space :

A pyramid SABCD is such that the base ABCD is a parallelogram.

Let’s call I, J, K the midpoints of the edges [SB], [SC] and [AB]

1) Show that the lines (IJ) and (AD) are parallel

2) Deduce from question 1) that the plane (SDK) and the line (IJ) are secant

3) Justify and construct the intersection of the planes (SKD) and (SBC)

4) Justify and construct the intersection of the line (IJ) with the plane (SKD)

Exercise 5:

Let ABCDEF, a right prism, I a point of ]DE[, J a point of ]DF[ and K, the center of the face BCFE of the prism. We are interested in the intersection of the planes (IJK) and (ABC).

^1st case : (IJ)//(EF)

1) Show that the intersection of (IJK) with (BCF) is parallel to (IJ). We will call $\Delta$ this intersection.

2) We call L the intersection of $\Delta$ with (EB) and M the intersection of D with (FC). Construct below the intersection of (IJK) with (ABC). We will only justify the existence of the additional points necessary for the construction or the use of the properties on parallelism.

^2nd case : (IJ) is not parallel to (EF). We will call N their point of intersection.

3) Without justification, construct below the intersection of (IJK) with (BCF) and then of (IJK) with (ABC).

Things to remember:

Skills to assimilate about geometry in space:

what to remember

Know the different solids of space (cube, rectangular parallelepiped, cylinder, right prism, cone of revolution and pyramid).
Know how to calculate the volume of a solid and represent it in space by means of its cavalier perspective or create its pattern;
Study the relative position of lines and planes in space.
These exercises are in accordance with the officialnational education programs.

In addition, you can consult the course on geometry in space and the relative position of lines and planes in space in second grade.

Cette publication est également disponible en : Français (French) العربية (Arabic)

Geometry in space: math exercises in 2nd grade corrected in PDF.