Triangle ABC

[nycmathguy]

Set 1.4
Question 26

See attachment.

Work out (a) through (d).

 

[MathLover1]

a)

A=(1,1)
B=(9,3)
C=(3,5)

the lenth of sides:
AB=sqrt((9-1)^2+(3-1)^2)=sqrt(8^2+2^2)=2sqrt(17)
AC=sqrt((3-1)^2+(5-1)^2)=sqrt(2^2+4^2)=2sqrt(5)
BC=sqrt((3-9)^2+(5-3)^2)=sqrt((-6)^2+2^2)=2sqrt(10)

perimeter=2sqrt(17)+2sqrt(5)+2sqrt(10)

perimeter=2(sqrt(17)+sqrt(5)+sqrt(10))

approximately, perimeter=19.04


b)

triangle formed by joining midpoints of the three sides

A=(1,1)
B=(9,3)
C=(3,5)

midpoint AB is ((1+9)/2,(1+3)/2)=(10/2,2)=(5,2)...... => A'=(5,2)
midpoint AC is ((1+3)/2,(1+5)/2)=(2,3)..........................=> B'=(2,3)
midpoint BC is ((9+3)/2,(3+5)/2)=(6,4)..........................=> C'=(6,4)

the length of sides:
A’B'=sqrt((2-5)^2+(3-2)^2)=3.16
A’C'=sqrt((6-5)^2+(4-2)^2)=2.24
B’C'=sqrt((6-2)^2+(4-3)^2)=4.12

perimeter=3.16+2.24+4.12
perimeter=9.52


c)

ratio of the perimeter in part (a) to the perimeter in part (b)

perimeter(a)/perimeter(b)=19.04/9.52=2

d)
The Midsegment Theorem

The Midsegment Theorem states that the midsegment connecting the midpoints of two sides of a triangle is parallel to the third side of the triangle, and the length of this midsegment is half the length of the third side.

 

[nycmathguy]

a)

A=(1,1)
B=(9,3)
C=(3,5)

the lenth of sides:
AB=sqrt((9-1)^2+(3-1)^2)=sqrt(8^2+2^2)=2sqrt(17)
AC=sqrt((3-1)^2+(5-1)^2)=sqrt(2^2+4^2)=2sqrt(5)
BC=sqrt((3-9)^2+(5-3)^2)=sqrt((-6)^2+2^2)=2sqrt(10)

perimeter=2sqrt(17)+2sqrt(5)+2sqrt(10)

perimeter=2(sqrt(17)+sqrt(5)+sqrt(10))

approximately, perimeter=19.04


b)

triangle formed by joining midpoints of the three sides

A=(1,1)
B=(9,3)
C=(3,5)

midpoint AB is ((1+9)/2,(1+3)/2)=(10/2,2)=(5,2)...... => A'=(5,2)
midpoint AC is ((1+3)/2,(1+5)/2)=(2,3)..........................=> B'=(2,3)
midpoint BC is ((9+3)/2,(3+5)/2)=(6,4)..........................=> C'=(6,4)

the length of sides:
A’B'=sqrt((2-5)^2+(3-2)^2)=3.16
A’C'=sqrt((6-5)^2+(4-2)^2)=2.24
B’C'=sqrt((6-2)^2+(4-3)^2)=4.12

perimeter=3.16+2.24+4.12
perimeter=9.52


c)

ratio of the perimeter in part (a) to the perimeter in part (b)

perimeter(a)/perimeter(b)=19.04/9.52=2

d)
The Midsegment Theorem

The Midsegment Theorem states that the midsegment connecting the midpoints of two sides of a triangle is parallel to the third side of the triangle, and the length of this midsegment is half the length of the third side.

Nicely-done and not too tedious.