Barycenter: answer key to senior math exercises in PDF.

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The answer key to the math exercises in senior year on the barycenter. Problems on the barycenter of weighted points. We will find the notions concerning the center of gravity of a triangle, the nature of a set of points, the concurrent lines.

Exercise 1:
It is up to you to make these constructions knowing that the barycenter is necessarily aligned with points A and B.

Exercise 7:
1. Describe the set of points M in the plane such that $\,\%7C5\vec{MA}+6\vec{MB}\%7C=22\,.$

Consider I barycenter of points (A,5) and (B,6)
$\,5\vec{MA}+6\vec{MB}=11\vec{MI}+5\vec{IA}+6\vec{IB}=11\vec{MI}+\vec{0}=11\vec{MI}\,.$
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 $\,\%7C-5\vec{MA}+8\vec{MB}\%7C=12\,.$

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

$\,-5\vec{MA}+8\vec{MB}=3\vec{MI}-5\vec{IA}+8\vec{IB}=3\vec{MI}+\vec{0}=3\vec{MI}\,.$

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 $\,\%7C5\vec{MA}-6\vec{MB}\%7C=\%7C7\vec{MA}-6\vec{MB}\%7C\,.$

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

$\,5\vec{MA}-6\vec{MB}=-\vec{MI}+5\vec{IA}-6\vec{IB}=-\vec{MI}+\vec{0}=-\vec{MI}\,.$

Also: $\,7\vec{MA}-6\vec{MB}=\vec{MJ}+7\vec{JA}-6\vec{JB}=-\vec{MJ}+\vec{0}=\vec{MJ}\,.$
so we are looking for the points M such that MI=MJ, the set is therefore the bisector of the segment [IJ].

Exercise 8:
Let R be an orthonormal reference frame of the plane .

1. Carry out the construction

2. We note $\,C_1$ the set of points M of the plane such that $\,\%7C4\vec{MA}+5\vec{MB}\%7C=45\,.$ .
$4\vec{MA}+5\vec{MB}=(4(3-x)+5(-1-x);4(4-y)+5(2-y))\,\\=(-9x+7;-9y+26).$ .

$\,\%7C4\vec{MA}+5\vec{MB}\%7C=\sqrt{(-9x+7)^2+(-9y+26)^2}\\=\sqrt{81%5B(x-\frac{7}{9})^2+(y-\frac{26}{9})^2%5D}.$

Determine the equation of the set $\,C_1$ .

The equation of this set is therefore :
$(x-\frac{7}{9})^2+(y-\frac{26}{9})^2=(\frac{45}{81})^2.$

2. Show that it is a mediator.

Exercise 9:

The answers to the exercises on the barycenter in high school.

After having consulted the answers to these exercises on the barycenter in senior year, you can return to the exercises in senior year.

Senior year exercises.

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The answers to the exercises on the barycenter in high school.

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