Worksheets on the barycenter in 11th grade with the use of the definition of the barycenter of n weighted points and properties of the barycenter such as associativity. All of these eleventh grade sheets have detailed answer keys so that students can review online.

Exercise 1 – Barycenter of weighted points
1. Construct the barycenter of the points {(A,1);(B,2)} given that AB = 6 cm .
2. Construct the barycenter of the points {(A,3);(B,-3)} given that AB = 8 cm .
3. Construct the barycenter of the points {(A,1);(B,-2)} given that AB = 4 cm .
4. Construct the barycenter of the points {(M,-3);(N,-2)} given that MN = 10 cm .

Exercise #2:
1. Describe the set of points M in the plane such that
2. Describe the set of points M in the plane such that
3. Describe the set of points M in the plane such that
4. Describe the set of points M in the plane such that

Exercise #3:
Let R be an orthonormal reference frame of the plane .
1. Construct the barycenter G of the points {(A,2);(B,3)} knowing that the coordinates, in R, of these points are A(3;4) and B(-1;2) .
2. We note the set of points M of the plane such that .
Determine the equation of the set .
2. We note the set of points M of the plane such that .
Determine the equation of the set .

Find a point location

ABC is an equilateral triangle of side 4 cm.
Determine the set of points M in the plane such that :
.

Exercise 4 – Determining a locus of points
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.
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 :

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

Exercise 5 – Exercise in a frame of reference
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?
3. Does the line (BG) pass through the origin of the reference frame? (Justify)

Exercise 6 – Aligning points
In triangle ABC, E is the middle of [AB] and G is the barycenter of (A,-2) (B,-2) (C,15).
Prove that G, C and E are aligned.

Exercise 7 – Classical Barycenter
ABCD is a quadrilateral and G is the barycenter of (A,1) (B,1) (C,3) (D,3).
Construct the point G. (Argue)

Exercise 8 – Isobarycenter and 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 – Physical Sciences

A balance consists of a mass M and a pan attached to the ends of a rod.
To weigh a mass m, the seller places a hook on the rod at a specific position.
This scale has the advantage for the trader not to handle several masses.

1. For each of the following cases, where should the hook G be attached to the segment [AB] to achieve balance?
(M = 2 kg)

These diagrams can be reproduced at any scale.

2. The point G is such that .

What is the mass m weighed? (M = 2 kg).

Exercise 10 – Determining the position of a barycenter

ABC is a triangle. G is the barycenter of (A; 2), (B; 1) and (C; 1).

The purpose of this exercise is to determine the precise position of the G-point.

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

Show that :

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

3. Conclude.

Exercise 11 – Construction and positioning

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 – Proving that points are aligned

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.

Exercise 13 – Confounded Barycenters

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).

Exercise 14 – Construction of barycenter in a triangle

ABC is a triangle.

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

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

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

Exercise 15 – Construction of a barycenter

ABCD is a quadrilateral and G is the barycenter of (A; 1), (B; 1), (C; 3) and (D; 3).

Construct the point G. Explain.

Exercise 16 – Set of points
ABCD is a square with center G and side 4 cm.
1. Calculate the length GA .
2. Reduce the sum ( using point G).
3. Determine and represent the set of points M such that :

4. Determine and represent the set of points M such that :
is collinear to .

Exercise 17 – Aligning points
In the triangle ABC, the point E is the middle of [AB]
and G is the barycenter of (A; -2) (B;-2) and (C;8).
1. Express E as the barycenter of A and B .
2. Prove that G, C and E are aligned.
3. Is C the middle of [EG]?

Exercise 18 – Equilateral triangle and parallel lines

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).

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

Exercise 19 – Center of gravity and intersecting lines
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 – Proving that lines intersect

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 – Proving that lines are parallel

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.

Exercise 22 – Barycenter and Locus

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 – A geometric place
[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. Then determine and represent the set of points M in the plane such that 2MA² + 3MB² = 245.

Exercise 24 – Set of points
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 – Associativity of the barycenter
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 – Barycenter and parameter
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].
1. Explain why Gm always exists and show that, when m describes , G
m
describes a line D that 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 benchmark ,
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 – Center of gravity
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 – Finding a set of points in the plane
ABCD is a square.

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

2. Represent this set E.

Exercise 29 – Square

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.

2. Reduce the writing of the following vectors:

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

3. Place K and justify.

Exercise 30 – Barycenter and point placement

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.

Exercise 31 – Isobarycenter, center of gravity and reference frame
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.

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

Exercise 32 – Set of points
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.

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 :

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

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