Exercise 111 Page 446

Practice makes perfect

a We can substitute the given coordinates of the vertices of the triangle into the distance formula to find the length of each side. We will also label each of the vertices with a letter to make our calculations easier.

Let's start with the length of XY.

XY=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

SubstitutePoints

Substitute ( 6,0) & ( 0,6)

XY=sqrt(( 0- 6)^2+( 6- 0)^2)

▼

Simplify right-hand side

SubTerms

Subtract terms

XY=sqrt((-6)^2+(6)^2)

CalcPow

Calculate power

XY=sqrt(36+36)

AddTerms

Add terms

XY=sqrt(72)

We can find the lengths of the other sides the same way.

Side	Expression	Length
XY	sqrt((0-6)^2+(6-0)^2)	XY=sqrt(72)
YZ	sqrt((6 - 0)^2+(6-6)^2)	YZ=6
ZX	sqrt((6-6)^2+(0-6)^2)	ZX=6

Notice that two of the lengths of the sides are the same. This means that △ XYZ is isosceles.

b Like in Part A, we can use the given coordinates of the vertices and the distance formula to find the length of each side. Let's also label each of the vertices with a letter.

We will start with the length of XY.

XY=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

SubstitutePoints

Substitute ( -3, 7) & ( -5, 2)

XY=sqrt(( -5-( -3))^2+( 2- 7)^2)

▼

Simplify right-hand side

SubNeg

a-(- b)=a+b

XY = sqrt((-5+3)^2+(2-7)^2)

AddSubTerms

Add and subtract terms

XY = sqrt((-2)^2+(-5)^2)

CalcPow

Calculate power

XY = sqrt(4+25)

AddTerms

Add terms

XY=sqrt(29)

We can find the lengths of the other sides the same way.

Side	Expression	Length
XY	sqrt((-5 - (-3))^2+(2-7)^2)	XY=sqrt(29)
YZ	sqrt((-1-(-5))^2+(2-2)^2)	YZ=4
ZX	sqrt((-3-(-1))^2+(7-2)^2)	ZX=sqrt(29)

Notice that two of the lengths of the sides are the same. This means that △ XYZ is isosceles.

c Like in Parts A and B, we will use the distance formula again to find the length of each side of the triangle. Let's also label each of the vertices with a letter.