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**answer key to the math exercises in 1st grade on the barycenter n weighted points**. Use the stability and associativity properties of the barycenter in first grade.

Exercise 1:

It is up to you to make these constructions knowing that the barycenter is necessarily aligned with points A and B.

Exercise 2:

1. Describe the set of points M in the plane such that

Consider I barycenter of points (A,5) and (B,6)

so the set corresponds to the circle of center I and radius 2 .

2. Describe the set of points M in the plane such that

Consider I barycenter of points (A,-5) and (B,8)

so the set corresponds to the circle of center I and radius 4 .

3. Describe the set of points M in the plane such that

Consider I barycenter of points (A,5) and (B,-6) and J barycenter of points (A,7) and (B;-6)

Also:

so we are looking for the points M such that MI=MJ, the set is therefore the bisector of the segment [IJ].

Exercise 3:

Let R be an orthonormal reference frame of the plane .

1. Carry out the construction

2. We note the set of points M of the plane such that .

.

Determine the equation of the set .

The equation of this set is therefore :

2. Show that it is a mediator.

Exercise 4:

Consider the point G, barycenter of (A,1); (B,1) and (C,2)

Consider the point K, barycenter of (B,1) and (C,3)

so it is the same as looking for the set of points in the plane such that :

**Conclusion: the set of points M is the bisector of the segment [MK]**

Exercise 7:

Let’s use the associativity of the barycenter

Let K be the barycenter of (A,1) (B,1) and L the barycenter of (C,3) (D,3)

The points K and L are isobarycenters so they are the midpoints of segments.

Moreover, these three barycenters exist because the sum of their masses is non-zero.

By associativity of the barycenter, G is the barycenter of (K,2) (L,6)

** To construct G, it is sufficient to place the point K in the middle of [AB] and L in the middle of [CD]. **

and

** Then it is enough to place the point K at the quarter of the segment [LK] starting from the point L. **

Exercise 8:

** Hint: use the associativity of the barycenter.**

ABCD is a quadrilateral.

We note G its isobarycenter.

**The purpose of this exercise is to clarify the position of G.**

1) Let I be the midpoint of [AB] and J the midpoint of [CD].

Show that G is the barycenter of I and J with coefficients to be specified.

2) Conclude and make a figure.

Exercise 9:

**1. For each of the following cases, where should the hook G be attached to the segment [AB] to achieve balance?**

**(M = 2 kg)**

** CASE 1: **

Let G be the barycenter of (A,2) and (B,3)

G exists because

** Conclusion: the hook must be placed at of [AB] starting from point A.**

** Case 2: **

Let G be the barycenter of (A,2) and (B,5)

G exists because

** Conclusion: the hook must be placed at of [AB] starting from point A.**

These diagrams can be reproduced at any scale.

Exercise 10:

**1. Let I be the middle of [BC].**

**Show that :**

** **

Using the associativity of the barycenter, we can state that

So we have :

or

(definition of the barycentre G)

so

**2. Deduce that G is the barycenter of A and I with coefficients to be specified.**

3. Conclude.

** Conclusion : G is the middle of the segment [AI].**

Exercise 11:

** Hint: G is the middle of [AI] and use the associativity of the barycenter.**

Consider a triangle ABC and designate by G the barycenter of (A; 1), (B; 4) and (C; – 3).

1. Construct the barycenter I of (B; 4) and (C; – 3).

2. Show that .

3. Deduce the position of G on (AI).

Exercise 12:

In triangle ABC, E is the middle of [AB] and G is the barycenter of (A; – 2), (B; – 2) and (C; 15).

**Show that G, C and E are aligned.**

First, the barycenter G exists because

We have:

and

**Let’s use the Chasles relation:**

or because E is the middle of [AB].

so we have the equality :

We conclude that the vectors and are collinear.

Moreover the point G belongs to these two vectors

therefore** the points G, C and E are aligned.**

Exercise 13:

B is the middle of [AC].

**Show that the barycenter of (A; 1) and (C; 3) is coincident with that of (B; 2) and (C; 2).**

Let and

We have:

or

and

IN SUMMARY:

and

(because B is the middle of [AC] ) and

thus:

** Conclusion: the points G and H are merged.**

Exercise 14:

ABC is a triangle.

**1. G is the barycenter of (A; 1), (B; 2) and (C; 3). Construct the point G. Explain.**

Let’s use the associativity of the barycenter.

G is the barycenter of (A,1) (I,5) with I the barycenter of (B,2) and (C,3)

**2. G ‘ is the barycenter of (A ; 1), (B ; 3) and (C ; – 3). Construct the point G ‘ . Explain.**

Let’s use the associativity of the barycenter,

G’ is the barycenter of (K,4) and (C,-3) with K being the barycenter of (A,1) and (B,3)

**3. Show that (AG’) is parallel to (BC).**

**Hint: show that the vectors and are collinear. **

Exercise 15:

First of all G exists because

Note and

Let’s use** the associativity of the barycenter**.

then

We place I which is the middle of [AB].

We place J in the middle of [CD]

then :

The segment [IJ] is divided into four equal parts

and we place the point G at three quarters of [IJ] starting from I.

Exercise 17:

Exercise 18:

Let ABC be an equilateral triangle of side 3 cm.

1) Locate, with justification, the barycenter Z of (A; 1), (B; 3) and (C; – 3).

z exists because

**2) Show that the lines (AZ) and (BC) are parallel.**

According to the above, .

These vectors are collinear, so we have (AZ) parallel to (BC).

Exercise 19:

**Hint: use the fact that G is the isobarycenter of the system {(A,1);(B,1);(C,1)}**

**and use the associativity of the barycenter.**

ABC is a triangle with center of gravity G.

I, J, M, N, R and S are the points defined by :

**Show that the lines (IS), (MR) and (NJ) are concurrent at G.**

Exercise 20:

** Hint: we can consider the barycenter G of (A; 5), (B; 2) and (C; – 3).**

ABC is a triangle.

Consider the barycenter A’ of (B; 2) and (C; – 3), the barycenter B ‘ of (A; 5) and (C; – 3)

and the barycenter C ‘ of (A; 5) and (B; 2).

Show that the lines (AA ‘), (BB ‘) and (CC ‘) are concurrent.

Exercise 21:

ABC is a triangle. Let G be the barycenter of (A; 1), (B; 3) and (C; – 3).

**Show that the lines (AG) and (BC) are parallel.**

First, the barycenter G exists because

We have:

Let’s use the Chasles relation:

The vectors and are thus collinear

**the lines (AG) and (BC) are parallel.**

Exercise 22:

1. Place the points A(1 ; 2), B( – 3 ; 4) and C( – 2 ; 5) in a reference frame.

Let G be the barycenter of the weighted points (A; 3), (B; 2) and (C; – 4).

2. What are the coordinates of G? Place G.

3. Does the line (BG) pass through the origin of the reference frame? Justify.

Exercise 23:

*[AB] *is a segment of length 10 cm and G bar {(A ; 2) , (B ; 3)}

1. Expand and reduce

2. Then prove that for any point M in the plane we have *2MA² + 3MB² = 5MG² + 120*

3. Determine then and **represent the set of points M of the plane** such *that 2MA² + 3MB² = 245*

Exercise 24:

Hint: use the definition and properties of the barycenter.

A, B and C are 3 points of the plane not aligned and k is any real number.

I bar { (B ;1), (C ;2)} and *G* the barycenter of*(A*, k),*(B*, 1- k) and*(C*, 2)

**1. **Express as a function of , and .

2. Simplify the expression obtained in 1. and deduce the set (E) of points G when k describes .

**3**. Graphically represent (E) in the case *AB = 5 cm, BC = 6 cm, AC = 5.5 cm*

Exercise 25:

Tip: Don’t complicate your life with hellish calculations, use cleverly

barycentric associativity.

A, B, C and D are four distinct points.

Let K be the barycenter of (A, 3) (B, 1), J the midpoint of [DC], G the center of gravity of BCD and I the midpoint of [AG].

Show that the points I, J and K are aligned.

Exercise 26:

**Hints: use the properties of the barycenter and associativity.**

ABC a triangle; to any real *m*, we associate the point G*
_{m}
* the barycenter of (A; 2); (B;

*m*) and (C; –

*m*).

We note O the middle of [BC].

Explain why _{Gm} always exists and show that, when *m* describes , G*
_{m}
* describes a line D which you will specify. >

2. a) Construct

_{G2}and

_{ G-2}. With AB= 4cm , AC = 3cm and BC = 6cm >

b) Assume

*m*is different from 2 and -2. >

Let G

*be a point of D distinct from A,*

_{m}_{G2}

_{ andG-2}. >

Prove that (BG

*) intersects (AC) at a point I and that (CG*

_{m}*) intersects (AB) at a point noted J. >*

_{m}3. in the frame , >

calculate the coordinates of I and J as a function of

*m*. >

Deduce that the points O, I and J are aligned. >

(We can use the analytical condition of collinearity of 2 vectors) >

Exercise 27:

**Hint: use the fact that the center of gravity is an isobarycenter and then the associativity of the barycenter.**

Let ABC be a triangle, A’ , B’ , and C’ the midpoints of the sides opposite A, B and C respectively, M a given point.

We note_{A1 },_{B1 }and_{C1 }the symmetrical of point M with respect to A’ , B’ , and C’ .

M’ is the barycenter of the points (A,1) (B,1) (C,1) and (M,-1)

1. Show that the lines (_{AA1}), (_{BB1}) and (_{CC1}) are concurrent at M ‘. >

2. Let G be the center of gravity of ABC. Show that M ‘ , M and G are aligned and specify the position of M ‘ on the line (MG). >

Exercise 28:

ABCD is a square.

1. What is the set E of points M of the plane such that :

Note

G exists because

because

** Conclusion: the set of points we are looking for is the circle with center G and radius .**

2. Represent this set E.

Exercise 29:

Let ABCD be a square and K the barycenter of the weighted points (A; 2), (B; – 1), (C; 2) and (D; 1).

Let I be the barycenter of the weighted points (A; 2) and (B; – 1), and J the barycenter of (C; 2) and (D; 1).

**1. Place I and J and justify.**

First of all the barycenters I and J exist because and .

** Conclusion : I is the symmetrical of point A with respect to B or A is the middle of [IB].**

** Conclusion: J is located at one third of [CD] starting from C.**

**2. Reduce the writing of the following vectors:**

and

Deduce that K is the barycenter of (I; 1) and (J; 3).

let’s take one of the two equalities:

so K is the barycenter of (I; 1) and (J; 3).

Exercise 30:

Let ABC be a triangle and G a point verifying :

**Is point G the barycenter of the weighted points (A; 5), (B; 1) and (C; 3)? Justify.**

**Let’s multiply this vector equality by – 1 .**

Exercise 31:

In a benchmark ,

1.Place the points A(2; 1), B( – 1; 5), C(5; 7) and G(1; ).

2. Determine the coordinates of the isobarycenter I of points B and C.

3. Determine the coordinates of the center of gravity H of triangle ABC.

** hint: use the fact that H is the barycenter of (A,1) (B,1) (C,1)**

4. Is there a real k such that G is barycenter of (A; 1) and (B; k)? Justify.

Exercise 32:

Let ABC be an isosceles triangle at A such that BC = 8 cm and BA = 5 cm. Let I be the middle of [BC].

1. Place the point F such that .

and show that F is the barycenter of the points A and B weighted by real numbers that we will determine.

**Hint: insert Point F into the vector and then transpose it into the first member.**

2. P being a point of the plane, reduce each of the following sums:

3. Determine and represent the set of points M of the plane verifying :

**Hint: consider the barycenter of (B,1) (C,1) and the barycenter of (A,-1) (B,2)**

4. Determine and represent the set of points M of the plane verifying :

## The answers to the exercises on the barycenter in 1st grade.

After consulting **the answers to these exercises on the barycenter of n weighted points in 1st grade**, you can return to the** exercises in 1st grade**.

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