Homework # 10

(1)

Homework # 10

The aim of this homework isto study the famous Euler line onanexample.

Consider an orthonormaloordinategraph

(O; ~

i

,~

^j

)

^with ^the ^points

A( − 3, 1)

^,

B (5, 1)

^and

C( − 2, 8)

^. ^We^all

A ^′

^the ^midpoint^of

[BC]

^and

B ^′

^the ^midpoint^of

[AC]

^.

1. For the following questions, you willneed tond informationon the internet or in

maths books.

(a) Whowas Euler? When did he live?

(b) What is the Euler lineof a triangle?

() Draw the triangle

ABC

ând îtsÊuler ^line.

2. (a) Compute the oordinates of

A ^′

^and

B ^′

^.

(b) Find out equationsof the mediansthrough

A

^and

B

ⁱⁿ^triangle

ABC

^.

() Dedue the oordinates of

G

^, ^the ^entroid ^of

ABC

^.

3. Consider the point

R (1 , 4)

^.

(a) Compute the lengths

RA

^,

RB

^and

RC

^.

(b) What is the point

R

ⁱⁿ ^the ^triangle

ABC

^?

We now admit that two lines

∆

^and

∆ ^′

^with ^respetive slope-interept equations

y = mx + p

^and

y = m ^′ x + p ^′

^are perpendiular if and onlyif

m × m ^′ = −1

^.

4. (a) What is the slopeof any lineperpendiular to

( BC )

^.

(b) Dedue the slope-interept equation of the altitude through

A

ⁱⁿ

ABC

^.

() Find out the simplestequation of the altitude through

C

ⁱⁿ^triangle

ABC

^.

(d) Dedue the oordinates of

H

^, ^the ^orthoenter ^of ^the ^triangle

ABC

^.

5. Use vetors tohek that the points

G

^,

R

^and

H

^are ^ollinear.

6. The Euler line passes through a fourth important point in the triangle. Find the

name and denition of this point.

Homework # 10

Homework # 10

(O; ~

,~

)

A( − 3, 1)

B (5, 1)

C( − 2, 8)

A ′

[BC]

B ′

[AC]

ABC

A ′

B ′

A

B

ABC

G

ABC

R (1 , 4)

RA

RB

RC

R

ABC

∆

∆ ′

y = mx + p

y = m ′ x + p ′

m × m ′ = −1

( BC )

A

ABC

C

ABC

H

ABC

G

R

H

A ^′

B ^′

A ^′

B ^′

∆ ^′

y = m ^′ x + p ^′

m × m ^′ = −1