1.2: Circles

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The Distance Formula

Delbert is hiking in the Santa Monica mountains, and he would like to know the distance from the Sycamore Canyon trail head, located at $12-\mathrm{C}$ on his map, to the Coyote Trail junction, located at $8-\mathrm{F}$, as shown below.

$Screen Shot 2022-09-09 at 4.52.39 PM.png$

Each interval on the map represents one kilometer. Delbert remembers the Pythagorean theorem, and uses the map coordinates to label the sides of a right triangle. The distance he wants is the hypotenuse of the triangle, so

\begin{aligned}
d^2 &= 4^2+3^2 = 16 + 9 = 25 \\
d&= \sqrt{25} = 5
\end{aligned}

The straight-line distance to Coyote junction is about 5 kilometers.

The formula for the distance between two points is obtained in the same way.

We first label a right triangle with points $P_1$ and $P_2$ on opposite ends of the hypotenuse. (See the figure at right.) The sides of the triangle have lengths $|x_2 - x_1|$ and $|y_2 - y_1|$. We can use the Pythagorean theorem to calculate the distance between $P_1$ and $P_2$:

$d^2 = (x_2-x_1)^2 + (y_2 - y_1)^2$

$Screen Shot 2022-09-09 at 5.01.08 PM.png$

Taking the (positive) square root of each side of this equation gives us the distance formula.

Distance Formula.

The distance $d$ between two points $P_1\left(x_1, y_1\right)$ and $P_2\left(x_2, y_2\right)$ is

\[d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} \nonumber\]

Example 1.31

Find the distance between (2, −1) and (4, 3).

Answer

We substitute $(2,-1)$ for $\left(x_1, y_1\right)$ and $(4,3)$ for $\left(x_2, y_2\right)$ in the distance formula to obtain

$Screen Shot 2022-09-09 at 5.08.49 PM.png$

\begin{aligned}
d &=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} \\
&=\sqrt{(4-2)^2+[3-(-1)]^2} \\
&=\sqrt{4+16}=\sqrt{20}=2 \sqrt{5}
\end{aligned}

The exact value of the distance, shown at right, is $2 \sqrt{5}$ units. We obtain the same answer if we use $(4,3)$ for $P_1$ and $(2,-1)$ for $P_2$:

\begin{aligned}
d &=\sqrt{(2-4)^2+[(-1)-3]^2} \\
&=\sqrt{4+16}=\sqrt{20}=2 \sqrt{5}
\end{aligned}

We can use a calculator to obtain an approximation for this value, and find

\[2 \sqrt{5} \approx 2(2.236)=4.472 \nonumber\]

Caution 1.32

In the previous example, the radical $\sqrt{4+16}$ cannot be simplified to $\sqrt{4}+\sqrt{16}$. (Do you remember why not?)

Checkpoint 1.33

a Find the distance between the points (−5, 3) and (3, −9).

b Plot the points on a Cartesian grid and show how the Pythagorean theorem is used to calculate the distance.

Answer: $4\sqrt{13}$

Equation for a Circle

A circle is the set of all points in a plane that lie at a given distance, called the radius, from a fixed point called the center. We can use the distance formula to find an equation for a circle.

The circle shown below has its center at the origin, (0, 0), and its radius is $r$.

$Screen Shot 2022-09-09 at 5.24.01 PM.png$

Now, the distance from the origin to any point $P(x, y)$ on the circle is $r$. Therefore,

$\sqrt{(x-0)^2+(y-0)^2} = r$

or, squaring both sides,

$(x-0)^2 + (y-0)^2 = r^2$

Because every point on the circle must satisfy this equation, we have found an equation for the circle.

Circle.

The equation for a circle of radius $r$ centered at the origin is

$x^2 + y^2 = r^2$

Example 1.34

Find two points on the circle $x^2+y^2=4$ with $y$-coordinate -1.

Answer

We substitute $x=-1$ into the equation for the circle, and solve for $y$.

\begin{aligned}
(-1^2)+y^2&=4 \\
y^2&=4-1=3 \\
y&=\pm\sqrt{3}
\end{aligned}

$Screen Shot 2022-09-09 at 5.30.15 PM.png$

The points are $(-1,\sqrt{3})$ and $(-1,\sqrt{3})$, as shown at right. Note that $\sqrt{3} \approx 1.732$.

Checkpoint 1.35

Find the coordinates of two points on the circle $x^2 + y^2 = 1$ with $y$-coordinate $\dfrac{1}{2}$.

Answer: $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right),\left(\frac{-\sqrt{3}}{2}, \frac{1}{2}\right)$

Definition 1.36 Unit Circle.

The circle in the exercise above, $x^2+y^2=1$, which is centered at the origin and has radius 1 unit, is called the unit circle.

Rational and Irrational Numbers

Every common fraction, such as $\dfrac{3}{4}$, can be written in many equivalent forms, including a decimal form. For example,

$\dfrac{3}{4}=\dfrac{6}{8}=\dfrac{75}{100}=0.75 \quad \text { and } \quad \dfrac{5}{11}=\dfrac{20}{44}=\dfrac{50}{110}=0 . \overline{45}$

where $0 . \overline{45}$ is the repeating decimal $0.45454545 \ldots$

Because the fraction bar denotes division, a fraction is a quotient of two integers, and we can calculate its decimal form by dividing the denominator into the numerator.

Definition 1.37 Raitonal Number.

Any number (including fractions) that can be written as a quotient of two integers $\frac{a}{b}$, where $b \neq 0$, is called a rational number.

The decimal form of a rational number is either a terminating decimal, such as $0.75$, or a repeating decimal, such as $0 . \overline{45}$. Thus, we can always write down an exact decimal equivalent for a rational number, although we may choose to round off a particularly long or unwieldy decimal. For example,

$\dfrac{3}{7}=0 . \overline{428571} \approx 0.43$

Caution 1.38

Note that $0.43$ is not exactly equal to $\frac{3}{7}$; it is an approximation for $\frac{3}{7}$, just as $0.33$ is an approximation for $\frac{1}{3}$. In our work, it will be important to distinguish between exact values and approximations.

Definition 1.39 Irrational Number.

An irrational number is one that cannot be written as a quotient of two integers $\frac{a}{b}$, where $b \neq 0$.

Examples of irrational numbers are $\sqrt{3}, \sqrt[3]{586}$, and $\pi$. The decimal form of an irrational number is nonterminating and nonrepeating. The first few digits of the examples mentioned are

\begin{aligned}
\sqrt{3} &=1.732050807568 \ldots \\
\sqrt[3]{586} &=8.368209391204 \ldots \\
\pi &=3.141592653589 \ldots
\end{aligned}

but none of these decimal forms ever ends. Thus, we cannot write down an exact decimal equivalent for an irrational number. The best we can do is give a decimal approximation, no matter how many digits we include.

Example 1.40

Which values are exact, and which are approximations?

a. $\dfrac{7}{32} \rightarrow 0.21875$

b. $\sqrt{8} \rightarrow 2.828427125$

c. $\sqrt{0.16} \rightarrow 0.4$

d. $\dfrac{\pi}{2} \rightarrow 1.570796327$

Answer

a. Because $\frac{7}{32}$ is a rational number, it has an exact decimal equivalent. Divide 7 by 32 to see that $\frac{7}{32}=0.21875$.

b. Because 8 is not a perfect square, $\sqrt{8}$ is irrational, so $2.828427125$ is not the exact value of $\sqrt{8}$.

c. $\sqrt{0.16}=\sqrt{\frac{16}{100}}=\frac{4}{10}=0.4$, so this value is exact.

d. Because $\frac{\pi}{2}$ is an irrational number, it has no decimal equivalent, so $1.570796327$ is an approximation.

Checkpoint 1.41

For which numbers can you give an exact decimal equivalent?

a. $\dfrac{\sqrt{3}}{2}$

b. $\dfrac{\sqrt{16}}{3}$

c. $2 \pi$

d. $\dfrac{25}{17}$

Answer: b, d

Circumference and Area

Recall from geometry that the circumference of a circle is proportional to its radius.

Circumference of a circle.

The circumference of a circle of radius $r$ is given by

$C=2 \pi r$

The number $\pi$ gives the ratio of the circumference of any circle to its diameter. It is an irrational number, $\pi \approx 3.14159$.

The length of a portion, or arc, of a circle, is called its arclength.

Example 1.42

Francine baked an apple pie with diameter 8 inches. If Delbert cuts himself a $60^{\circ}$ wedge, what is the arclength of the curved edge?

Answer

The radius of the pie is 4 inches, so its circumference is $2 \pi(4)$ inches.

A $60^{\circ}$ wedge, shown below, is $\frac{1}{6}$ of the entire pie, so its edge is $\frac{1}{6}$ of the circumference. The exact length of the arc is thus

$Screen Shot 2022-09-09 at 6.29.53 PM.png$

\[\dfrac{1}{6}(8 \pi)=\dfrac{4 \pi}{3} \text { inches. } \nonumber\]

Using a calculator, we find that $\frac{4 \pi}{3}=4.19$ rounded to two decimal places, so the length of the curved edge is between $4$ and $4 \frac{1}{2}$ inches.

Checkpoint 1.43

What is the arclength of the curved edge of a $60^{\circ}$ wedge cut from a blueberry pie of diameter 10 inches?

Answer: $\dfrac{5 \pi}{3} \approx 5.24$ inches

The area of a circle is proportional to the square of its radius.

Area of a Circle.

The area of a circle of radius $r$ is given by

$A=\pi r^2$

A portion of a circle shaped like a pie-shaped wedge is called a sector.

Example 1.44

What is the area of Delbert’s slice of apple pie in the previous example?

Answer

As we saw in the previous example, Delbert's sector of the pie is $\frac{1}{6}$ of the entire pie, so its area $\frac{1}{6}$ of the area of the whole pie, or

$\dfrac{1}{6} \pi\left(4^2\right)=\dfrac{8 \pi}{3}$ square inches

The area of the wedge is $\dfrac{8 \pi}{3}$, or about $8.34$ square inches.

Checkpoint 1.45

What is the area of a $60^{\circ}$ wedge cut from a blueberry pie of diameter 10 inches?

Answer: $\dfrac{25 \pi}{6} \approx 13.09$ square inches

Review the following skills you will need for this section.

Algebra Refresher 1.4

True of False.

1. $\sqrt{a^2+b^2}=a+b$

2. $\sqrt{36+64}=6+8$

3. $\sqrt{16 x^4}=4 x^2$

4. $\sqrt{2 x} \sqrt{3 y}=\sqrt{6 x y}$

5. $\sqrt{5 x}+\sqrt{3 x}=\sqrt{8 x}$

6. $\sqrt{4+N}=2+\sqrt{N}$

7. $\sqrt{\dfrac{x}{4}}=\dfrac{\sqrt{x}}{2}$

8. $\sqrt{\dfrac{3}{2}}=\dfrac{\sqrt{6}}{2}$

Algebra Refresher Answers

1 False

2 False

3 True

4 True

5 False

6 False

7 True

8 True

Section 1.3 Summary

Vocabulary

Look up the definitions of new terms in the Glossary.

• Rational number

• Irrational number

• Repeating decimal

• Circle

• Radius

• Unit circle

• Circumference

• Arclength