## Math 11 Chapter 2 Lesson 5: Parallel projection. Representation of a space figure

## 1. Summary of theory

### 1.1. Parallel projection

– For plane \(\left( \alpha \right)\) and straight line \(\Delta \) cut \(\left( \alpha \right)\). For each point M in space, the line passes through M and is parallel or coincident with \(\Delta \) will cut \(\left( \alpha \right)\) at the specified point M’.

– The point M’ is called the parallel projection of the point M on the plane . \(\left( \alpha \right)\) in the direction \(\Delta \).

– Plane \(\left( \alpha \right)\) called the projection plane. Direction \(\Delta \) called projection.

– The placement corresponding to each point M in space to its projection M’ on the plane \(\left( \alpha \right)\) is called the up parallel projection \(\left( \alpha \right)\) in the direction \(\Delta \).

**– Attention:** If a line has the same direction as the projection, the projection of that line is a point.

### 1.2. Properties of parallel projection

**– Theorem 1:**

+ Parallel projection turns three collinear points into three collinear points and does not change the order of those three points.

+ Parallel projection turns lines into lines, rays into rays, and segments into segments.

+ Parallel projection turns two parallel lines into two parallel or overlapping lines.

+ Parallel projection does not change the ratio of the lengths of two line segments lying on two parallel lines or lying on the same line.

### 1.3. The representation of a spatial figure on the plane

– The representation of a shape H in space is a parallel projection of the figure H on a plane in a certain projection direction or a shape congruent to that projection.

– Representations of common shapes

+ Triangle: Any triangle can always be considered a representation of a triangle of a given arbitrary shape (it can be an equilateral triangle, isosceles triangle, right triangle, …).

+ Parallelogram: Any parallelogram can always be considered as a representation of a parallelogram of a given arbitrary shape (can be a parallelogram, square, rhombus, rectangle, etc.) …).

+ Trapezoid: Any trapezoid can always be considered a representation of a trapezoid of a given arbitrary shape, as long as the ratio of the lengths of the two bases of the representation must be equal to the ratio of the lengths of the two bases. of the original trapezoid.

+ Circle: Use an ellipse to represent a circle.

## 2. Illustrated exercise

**Lesson 1: **Let the pyramid S.ABC have the sides are equilateral triangles. Let M be the midpoint of BC, and N the point on the side SA such that \(\frac{{SN}}{{NA}} = \frac{1}{2}\). Find the projection of N through the bilateral projection SM projection plane (ABC).

**Solution guide:**

We have triangles SAB, SAC, SBC which are equilateral triangles. The triangle ABC is also an equilateral triangle.

Let G be the centroid of triangle ABC.

We have \(\frac{{SN}}{{NA}} = \frac{{MG}}{{GA}} = \frac{1}{2} \Rightarrow \) NG // SM.

So G is a parallel projection in the SM direction of the upper N (ABC).

**Lesson 2:** Let ABC.A’B’C’ prism. Let G be the centroid of triangle ABC. Through bilateral projection AA’ projection plane (A’B’C’) turns G into G’. Prove that G’ is the centroid of triangle A’B’C’.

**Solution guide:**

Let M be the mid point of AB.

Bilateral projection AA’ turns C into C’, turns M into M’.

We have G as the centroid of triangle ABC. It follows that C, M, G are collinear and \(\frac{{CG}}{{CM}} = \frac{2}{3}\).

Therefore C’, G’, M’ are collinear and \(\frac{{C’G’}}{{C’M’}} = \frac{2}{3}\).

On the other hand, M is the midpoint of AC, so M’ is the midpoint of A’B’.

So G’ is the centroid of triangle A’B’C’.

## 3. Practice

### 3.1. Essay exercises

** Lesson 1:** Given the box ABCD.A’B’C’D’, find the images of points C, D’.

a) Through the bilateral projection AB the projection plane (BCC’B’).

b) Through bilateral projection A’B projection plane (CDD’C’).

**Lesson 2:** Given a pyramid S.ABCD whose base is a parallelogram, O is the center of the base. On the edge SB, SD takes the points M, N respectively, so that SB = 3MB, ND = 2 SN. The projection of M, N through the bilateral projection SO projection plane (ABCD) is P, Q respectively. Calculate the ratio \(\frac{{BP}}{{QD}}\).

**Lesson 3:** Given the pyramid S.ABCD, the base is a parallelogram. Let O be the center of the base, take M and N as the images of points A and D, respectively, through the parallel projection SO on the plane (SBC). Then what is the quadrilateral BCNM? What condition does the pyramid S.ABCD add to make quadrilateral BCNM a square?

### 3.2. Multiple choice exercises

**Lesson 1:** Suppose triangle ABC is an equilateral triangle. The representation of the circumcenter of an equilateral triangle is:

A. The intersection of the two medians of triangle ABC.

B. The intersection of two perpendicular bisectors of triangle ABC.

C. The intersection of the two altitudes of triangle ABC.

D. The intersection of two bisectors of triangle ABC.

**Lesson 2:** Let the pyramid S.ABCD whose base ABCD is the great trapezoid CD. Which of the following planes can make the projection plane in the projection parallel to the CD direction?

A. (SCD).

B. (ABCD).

C. (SAB).

D. (SBD).

**Lesson 3:** Given the prism ABC.A’B’C’, through the bilateral projection AA’, the projection plane (ABC) turns I into I’. where I is the midpoint of A’C’. Choose the correct statement?

A. I’ is the midpoint of AB.

B. I’ is the midpoint of AC.

C. I’ is the midpoint of BC.

D. All three answers above are wrong.

**Lesson 4:** Given a pyramid S.ABCD whose base is a parallelogram, on the side SA take a point M such that \(MA=3MS\). Let O be the center of the base, through the bilateral projection MO projection plane (ABCD) turns point S into point N. Calculate the ratio \(\frac{{CN}}{{CA}}\).

A. \(\frac{1}{3}\).

B. \(\frac{3}{4}\).

C. \(\frac{1}{6}\).

D. \(\frac{2}{3}\).

**Lesson 5:** Let tetrahedron ABCD and M be any point in the region of triangle BCD. Let B’, C’, D’ be the parallel projections of M in the directions AB, AC, AD on the faces (ACD), (ABD), (ABC) respectively. Calculate \(\frac{{MB’}}{{AB}} + \frac{{MC’}}{{AC}} + \frac{{MD’}}{{AD}}\)?

A. \(\frac{1}{9}\).

B. \(\frac{1}{3}\).

C. 1.

D. 3.

## 4. Conclusion

Through this lesson, you should be able to understand the following:

– The concept of parallel projection; Conceptual representation of a spatial figure.

– Determine: projection direction, projection plane in a parallel projection. Construct the image of a point, a line segment, a triangle, a circle through a parallel projection.

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