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**answer key to the math exercises in 1st grade in PDF on the scalar product in the plane**. Use the bilinearity properties of the scalar product and demonstrate that two vectors are orthogonal or collinear. Apply the Chasles relation to vectors in the first grade.

Exercise 1:

a. AB=3 , AC=5 and .

b. AB=1 , AC=4 and .

c. AB=4 , AC=7 and .

d. AB=2 , AC=2 and .

Exercise 2:

Calculate knowing that :

a.

Exercise 4:

Let ABCD be a square and I a point of [AB].

Let H be the orthogonal project of A onto [ID].

because (IA) is perpendicular to (AD).

Exercise 5:

Let ABC be an equilateral triangle of side 1.

Let H be the orthogonal project of A onto (BC).

and .

Exercise 6:

The purpose of the problem is to show that the lines (CQ) and (PR) are perpendicular.

1. Justify that: .

( according to **the relation of Chasles**)

2. Deduce that the lines (CQ) and (PR) are perpendicular.

**Indication:**

Show that

Exercise 7:

We have and and . = -1

1) Calculate and

2) Calculate ( + ) . (2-3)

Exercise 8:

Prove that whatever the point M of the plane, we have the equality :

and I is the middle of [AB] so

Exercise 10:

Exercise 11:

Hint: **create a marker in the parallelogram**.

ABCD is a parallelogram with AB = 4, AD = 5 and AC = 7.

1.Calculate.

2. Deduct BD.

Exercise 12:

MNPQ is a square with MN = 6. I is the center of the square.

Calculate the following scalar products:

1.

2.

3.

4.

Exercise 13:

ABC is a triangle in which AB = 2 and AC = 3.

In addition

**Is this triangle rectangular? If yes, specify in which summit.**

**Conclusion: the triangle is not rectangular because the cosine is different from zero.**

Exercise 14:

ABC is an equilateral triangle of side 5 cm. I is the middle of [BC].

Calculate the following scalar products:

1. .

2.

3.

(Ai) is a median but as the triangle is equilateral, it is also a height

so these two vectors are orthogonal and therefore their scalar product is zero.

Exercise 15:

In an orthonormal reference frame

we consider the following points: A (2; 1), B (7; 2) and C (3; 4).

*All of the following questions are independent and unrelated.*

**1. Compute the coordinates of the barycenter G of (A; 3), (B; 2) and (C; – 4).**

First of all abrod the barycenter exists because

2. Determine a Cartesian equation of the perpendicular bisector of [BC].

a direction vector of [BC] is

so a normal vector of the median is :

so a Cartesian equation is of the type :

and the midpoint I of [BC] belongs to the perpendicular bisector : so

We obtain:

** Conclusion: a Cartesian equation of the median is .**

3. Calculate .

**4. Is the angle right?**

No since the scalar product is not zero.

Exercise 16:

Exercise 17:

Exercise 18:

Exercise 19:

Knowing that the vectors and are such that , and .

Calculate the following scalar products:

**1.**

2.

Exercise 20:

Under which condition on points A, B and C do we have :

We have:

and

Therefore, it is necessary that :

so

that **the point A belongs to the line (BC) deprived of the segment [BC].**

Exercise 21:

1.

Let I be the middle of [AB] and therefore the isobarycenter of [AB].

using the properties of the scalar product :

> 1

** Conclusion: this is the circle with center I and radius dm .**

2.

Exercise 22:

[AB] is a segment of middle I and AB = 2 cm.

1. Show that for any point M in the plane :

We have:

because I is in the middle of [AB] so

Exercise 23:

Prove that :

1. .

2. .

3. What is the link with the rhombus, the parallelogram?

**Make a figure…**

4. Show that :

(because the scalar product is symmetrical)

5. Deduce that a parallelogram has its diagonals perpendicular if and only if its sides are equal.

**Make a figure….**

Exercise 24:

Indication:

**The equation of a circle is **

**with center of the circle of radius R.**

In an orthonormal reference frame, we give a point .

1. Determine the equation of the circle (C) with center and radius R = 5.

2. Prove that the point A( – 2 ; 0) is a point of the circle (C).

3. Determine a Cartesian equation of the tangent at A to the circle (C).

Exercise 26:

We place ourselves in an orthonormal reference frame

Consider a triangle ABC with A (-1; 2), B (3; 1) and C (2; 4).

**1. Determine an equation of the perpendicular bisector of the segment [AB].**

Let’s determine the coordinates of the vector .

Let’s determine the coordinates of I, the middle of the segment [AB]:

The coordinates of a director vector of the median is a normal vector to the vector

so

Any point M(x,y) belongs to the bisector of [AB]

if and only if :

(k non-zero)

so

so the reduced equation of the perpendicular bisector of [AB] is :

2. Determine an equation of the height from A in triangle ABC.

Let’s determine the coordinates of the vector :

Let H be the orthogonal project of A onto (BC).

The line (AH) is therefore the height from the vertex A.

Let M(x,y) belong to (AH)

if and only if :

is a reduced equation of the height from A.

Exercise:

We place ourselves in an orthonormal reference frame .

1. Determine the equation of the circle with center tangent to the line (D) of equation :

**Indication: **

we recall that the distance between a point and a line (D) of equation ax + by + c = 0 is

given by the formula :

Let’s determine the distance between the tangent and the center of the circle, this will be the radius of the circle.

The equation of the circle is :

Exercise 27:

We place ourselves in an orthonormal reference frame .

Consider whether the following equations are equations of a circle and, if so, specify the center and radius of the circle.

1.

It is a circle with center point I ( 1 ; 3 ) and radius .

2.

**This is not the equation of a circle.**

Exercise 28:

** Hint: use the generalized Pythagorean formula in any triangle.**

ABC is a triangle and I is the middle of [BC].

We give: BC = 4, AI = 3 and .

**Calculate:**

1.

2.

3.

4.

## The answers to the exercises on the scalar product in the plane in 1ère.

After having consulted **the answers to these exercises on the scalar product in the plane in 1st grade**, you can return to the **exercises in 1st grade. **

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