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Big Ideas Math Geometry, 2014 View details

Chapter Test

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Exercise 16 Page 587

Practice makes perfect

Let's make a diagram describing the situation.

Diagram of circular skid marks, three radii, and a chord highlighted

First we will find the radius of the circle. After that we will explain why it works.

Finding the Radius

The two segments that intersect at C are perpendicular. Using this knowledge, we will now consider the triangle MCB.

Since we have a right triangle we can use the Pythagorean Theorem. a^2+b^2=c^2 In this formula, we will substitute (r-80) and 160 for a and b, respectively, and r for c. By solving the resulting equation we can find the circle's radius, r.

a^2+b^2=c^2

SubstituteValues

Substitute values

(r-80)^2+ 160^2=r^2

▼

Solve for r

ExpandNegPerfectSquare

(a-b)^2=a^2-2ab+b^2

r^2-2r(80)+80^2+160^2=r^2

SubEqn

LHS-r^2=RHS-r^2

- 2r(80)+80^2+160^2=0

CalcPowProd

Calculate power and product

- 160r+6400+25 600=0

AddTerms

Add terms

- 160r+32 600=0

SubEqn

LHS-32 000=RHS-32 000

- 160r=- 32 000

DivEqn

.LHS /-160.=.RHS /-160.

r= 200

Thus, the circle has a radius of 200 feet.

Why It Works

The segment AB is a chord in the circle. A perpendicular chord which bisects AB is, by the Perpendicular Chord Bisector Converse, a diameter in the circle.

Diagram showing a circle, its diameter, and a chord bisected by the diameter

The diameter DP passes through the circle's midpoint, M. Therefore, DM is the circle's radius.

We have been given a formula that in this situation can be used to estimate the speed the car had. S=3.87sqrt(f r) The coefficient of friction, f, has been estimated to 0.7. From Part A we know that the radius of the circle, r, is 200 feet. Let's substitute these values into the formula and calculate the car's speed.

S=3.87sqrt(fr)

SubstituteII

f= 0.7, r= 200

S=3.87sqrt(0.7( 200))

▼

Simplify right-hand side

UseCalc

Use a calculator

S=45.79045...

RoundInt

Round to nearest integer

S≈ 46