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
m
be a point of D distinct from A, G2 andG-2. >
Prove that (BG
m
) intersects (AC) at a point I and that (CG
m
) intersects (AB) at a point noted J. >
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 noteA1 ,B1 andC1 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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