Exercise 39 Page 26

Practice makes perfect

a The distance around the park is the sum of the lengths of each line segment of the triangle. Each unit represents 10 yards. The horizontal and vertical line segments are 70 yards and 40 yards, respectively.

The park is a right triangle, which means that we can use the Pythagorean Theorem to find length of PR.

PR^2=40^2+70^2

SqrtEqn

sqrt(LHS)=sqrt(RHS)

PR=sqrt(70^2+40^2)

CalcPow

Calculate power

PR=sqrt(4900+1600)

AddTerms

Add terms

PR=sqrt(6500)

UseCalc

Use a calculator

PR=80.6225...

The third side length is about 81 yards.

Now we can add the three lengths to find the total distance around the park. 70+40+81=191yards

b To find the length of the new path, we should start by finding the midpoint of P and R. The coordinates of the endpoints are (10,50) and (80,10).

Now, we can use the Midpoint Formula to find the midpoint.

(x_M,y_M)=(x_1+x_2/2,y_1+y_2/2)

SubstitutePoints

Substitute ( 10,50) & ( 80,10)

(x_M,y_M)=(10+ 80/2,50+ 10/2)

AddTerms

Add terms

(x_M,y_M)=(90/2,60/2)

CalcQuot

Calculate quotient

(x_M,y_M)=(45,30)

The midpoint is found at M(45,30).

Next, to find the length of QM, we need to use the coordinates of Q(10,10) and M(45,30) in the Distance Formula.

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

SubstitutePoints

Substitute ( 45,30) & ( 10,10)

QM = sqrt(( 45- 10)^2 + ( 30- 10)^2)

▼

Evaluate

SubTerms

Subtract terms

QM=sqrt(35^2+20^2)

UseCalc

Use a calculator

QM=40.31128...

RoundInt

Round to nearest integer

QM≈ 40

The length of QM is about 40 yards.

c The time it takes for the man to complete a run depends on the distance and his speed. Let's make a schematic diagram of the runner's path.

The lengths of PQ and RQ were determined in Part A. They are 40 yards and 70 yards.

Since PQ and QP describe the same line segment, QP is also 40 yards.

In Part B, we found the length of QM to be 40 yards too.

M is the midpoint of P and R, meaning it divides PR into two congruent line segments. We know, from previous calculations, that PR≈ 81 yards, so MR is 81/2=40.5 yards. This gives us the following diagram.

The total distance ran is the sum of all of the relevant lengths. 40+40+40.5+70+40=230.5 yards We know that his average speed was 150 yards per minute. Therefore, the run takes 230.5yards/150 yards per minute≈ 1.5 minutes.

Subchapter links

Communicate Your Answer p.19

Monitoring Progress p.20-23

Exercises p.24-26

Exercise 39 Page 26

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