Exercise 10 Page 57

We are asked to calculate the midpoint of two points and the distance between them. Let's do each step separately.

Calculating the Midpoint

The coordinates of the given endpoints are (-2,4) and (3,9). Let's use these to find the coordinates of the midpoint.

M(x_1+x_2/2,y_1+y_2/2)

SubstitutePoints

Substitute ( -2,4) & ( 3,9)

M(-2+ 3/2,4+ 9/2)

AddTerms

Add terms

M(1/2,13/2)

The midpoint is located at ( 12, 132). We can see that on the graph.

Calculating the Distance

To determine the distance between two points we can use the Distance Formula. Recall that the coordinates of the given points are S(-2,4) and T(3,9).

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

SubstitutePoints

Substitute ( -2,4) & ( 3,9)

ST=sqrt(( 3-( -2))^2+( 9- 4)^2)

SubNeg

a-(- b)=a+b

ST=sqrt((3+2)^2+(9-4)^2)

AddSubTerms

Add and subtract terms

ST=sqrt(5^2+5^2)

CalcPow

Calculate power

ST=sqrt(25+25)

AddTerms

Add terms

ST=sqrt(50)

▼

Simplify right-hand side

SplitIntoFactors

Split into factors

ST=sqrt(25*2)

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

ST=sqrt(25)*sqrt(2)

CalcRoot

Calculate root

ST=5sqrt(2)

The distance between S and T is 5sqrt(2). We can use a calculator to approximate its value. 5sqrt(2) ≈ 7.1