Chapter 1: Triangles and Circles
True or False?
1. [latex]\sqrt{a^{2}+b^{2}}=a+b[/latex]
2. [latex]\sqrt{36+64}=6+8[/latex]
3. [latex]\sqrt{16x^{4}}=4x^{2}[/latex]
4. [latex]\sqrt{2x}\sqrt{3y}=\sqrt{6xy}[/latex]
5. [latex]\sqrt{5x}+\sqrt{3x}=\sqrt{8x}[/latex]
6. [latex]\sqrt{4+N}=2+\sqrt{N}[/latex]
7. [latex]\sqrt{\frac{x}{4}}=\frac{\sqrt{x}}{2}[/latex]
8. [latex]\sqrt{\frac{3}{2}}=\frac{\sqrt{6}}{2}[/latex]
Learning Objectives
Delbert is hiking in the Santa Monica Mountains, and he would like to know the distance from the Sycamore Canyon trailhead, located at [latex]12[/latex]-C on his map, to the Coyote Trail junction, located at [latex]8[/latex]-F, as shown below.
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
[latex]d^{2} = 4^{2} + 3^{2} = 16 + 9 =25 \\ d = \sqrt{25} = 5[/latex]
The straight-line distance to Coyote junction is about [latex]5[/latex] kilometers.
The formula for the distance between two points is obtained in the same way.
We first label a right triangle with points [latex]P_{1}[/latex] and [latex]P_{2}[/latex] on opposite ends of the hypotenuse. (See the figure at right.) The sides of the triangle have lengths [latex]\lvert{x_2-x_1}\rvert[/latex] and [latex]\lvert{y_2-y_1}\rvert[/latex]. We can use the Pythagorean theorem to calculate the distance between [latex]P_{1}[/latex] and [latex]P_{2}[/latex]
[latex]d^{2} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}[/latex]
Taking the (positive) square root of each side of this equation gives us the distance formula.
Distance Formula
The distance [latex]d[/latex] between two points [latex]P_{1}(x_{1}, y_{1})[/latex] and [latex]P_{2}(x_{2}, y_{2})[/latex] is
[latex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/latex]
Example 1.3.1.
Find the distance between [latex](2,-1)[/latex] and [latex](4,3)[/latex]
We substitute [latex](2,-1)[/latex] for ([latex]x_{1}[/latex], [latex]y_{1}[/latex]) and [latex](4,3)[/latex] for ([latex]x_{2}[/latex], [latex]y_{2}[/latex]) in the distance formula to obtain
[latex]d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}} \\ = \sqrt{(4-2)^{2}+[3-(-1)]^{2}} \\ = \sqrt{4+16} = \sqrt{20} = 2\sqrt{5}[/latex]
The exact value of the distance, shown at right, is [latex]2\sqrt{5}[/latex] units. We obtain the same answer if we use [latex](4,3)[/latex]for [latex]P_{1}[/latex]and [latex](2,-1)[/latex] for [latex]P_{2}[/latex]
[latex]d = \sqrt{(2-4)^{2}+[(-1)-3]^{2}} \\ = \sqrt{4+16} = \sqrt{20} = 2\sqrt{5}[/latex]
We can use a calculator to obtain an approximation for this value, and find
[latex] 2\sqrt{5}\approx 2(2.236)=4.472[/latex]
In the previous example, the radical [latex]\sqrt{4+16}[/latex] cannot be simplified to [latex]\sqrt{4}+\sqrt{16}[/latex]. (Do you remember why not?)
[latex]4\sqrt{13}[/latex]
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, [latex](0,0)[/latex] and its radius is [latex]r[/latex].
Now, the distance from the origin to any point [latex]P(x,y)[/latex] on the circle is [latex]r[/latex]. Therefore,
[latex]\sqrt{(x-0)^{2}+(y-0)^{2}} = r[/latex]
or, squaring both sides,
[latex](x-0)^{2}+(y-0)^{2}=r^2[/latex]
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 [latex]r[/latex] centered at the origin is
[latex]x^{2}+y^{2}=r^{2}[/latex]
Example 1.34.
Find two points on the circle [latex]x^{2}+y^{2}=4[/latex] with x-coordinate [latex]-1[/latex].
We substitute [latex]x=-1[/latex] into the equation for the circle, and solve for [latex]y[/latex].
[latex](-1)^{2}+y^{2} = 4 \\ y^{2} = 4-1 = 3 \\ y = \pm \sqrt{3}[/latex]
The points are [latex](-1, \sqrt{3})[/latex] and [latex](-1,-\sqrt{3})[/latex] as shown at right. Note that [latex]\sqrt{3} \approx 1.732[/latex].
Find the coordinates of two points on the circle [latex]x^{2}+y^{2}=1[/latex] with [latex]y[/latex]-coordinate [latex]\frac{1}{2}[/latex]
[latex]\bigg( \frac{\sqrt{3}}{2},\frac{1}{2}\bigg)[/latex], [latex]\bigg( \frac{-\sqrt{3}}{2},\frac{1}{2} \bigg)[/latex]
Definition 1.36. Unit Circle
The circle in the exercise above, [latex]x^{2}+y^{2}=1[/latex], which is centered at the origin and has radius [latex]1[/latex] unit, is called the unit circle.
Every common fraction, such as [latex]\frac{3}{4}[/latex], can be written in many equivalent forms, including a decimal form. For example,
[latex]\frac{3}{4}=\frac{6}{8}=\frac{75}{100}=0.75 ~~~and~~~\frac{5}{11}=\frac{20}{44}=\frac{50}{110}=0.\overline{45}[/latex]
where [latex]0.\overline{45}[/latex] is the repeating decimal [latex]0.45454545....[/latex]
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. Rational Number
Any number (including fractions) that can be written as a quotient of two integers [latex]\frac{a}{b},~~\text{where}~~ b\neq 0[/latex] is called a rational number.
The decimal form of a rational number is either a terminating decimal, such as [latex]0.75[/latex] or a repeating decimal, such as [latex]0.\overline{45}[/latex]. 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,
[latex]\frac{3}{7}=0.\overline{428571}\approx 0.43[/latex]
Note that [latex]0.43[/latex] is not exactly equal to [latex]\frac{3}{7}[/latex]; it is an approximation for [latex]\frac{3}{7}[/latex] just as [latex]0.33[/latex] is an approximation for [latex]\frac{1}{3}[/latex]. 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 [latex]\frac{a}{b}[/latex] where [latex]b \neq 0[/latex].
Examples of irrational numbers are [latex]\sqrt{3}[/latex], [latex]\sqrt{5}[/latex], and [latex]\pi[/latex]. The decimal form of an irrational number is nonterminating and nonrepeating. The first few digits of the examples mentioned are
[latex]\sqrt{3} = 1.732050807568...\\ \sqrt{5} = 2.236067977..\\ \pi = 3.141592653589...[/latex]
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. Because [latex]\frac{7}{32}[/latex] is a rational number, it has an exact decimal equivalent. Divide [latex]7[/latex] by [latex]32[/latex] to see that [latex]\frac{7}{32}=0.21875[/latex].
b. Because [latex]8[/latex] is not a perfect square, ([latex]\sqrt{8}[/latex]) is irrational, so [latex]2.828427125[/latex] is not the exact value of ([latex]\sqrt{8}[/latex]).
c. [latex]\sqrt{0.16}=\sqrt{\frac{16}{100}}=\frac{4}{10}=0.4[/latex], so this value is exact.
d. Because ([latex]\frac{\pi}{2}[/latex]) is an irrational number, it has no decimal equivalent, so (1.570796327) is an approximation.
For which numbers can you give an exact decimal equivalent?
b, d
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 [latex]r[/latex] is given by
[latex]C=2 \pi r[/latex]
The number [latex]\pi[/latex] gives the ratio of the circumference of any circle to its diameter. It is an irrational number, [latex]\pi \approx 3.14159[/latex].
The length of a portion, or arc, of a circle is called its arclength.
Example 1.42.
Francine baked an apple pie with diameter [latex]8[/latex] inches. If Delbert cuts himself a [latex]60°[/latex] wedge, what is the arclength of the curved edge?
Solution.
The radius of the pie is [latex]4[/latex] inches, so its circumference is [latex]2 \pi 4[/latex] inches.
A [latex]60°[/latex] wedge, shown at right, is [latex]\frac{1}{6}[/latex] of the entire pie, so its edge is [latex]\frac{1}{6}[/latex] of the circumference. The exact length of the arc is thus
[latex]\frac{1}{6} (8 \pi) = \frac{4 \pi}{3} \text{ inches.}[/latex]
Using a calculator, we find that [latex]\frac{4 \pi}{3}=4.19[/latex] rounded to two decimal places, so the length of the curved edge is between [latex]4[/latex] and [latex]4\bigg( \frac{1}{2} \bigg)[/latex] inches.
What is the arclength of the curved edge of a [latex]60°[/latex] wedge cut from a blueberry pie of diameter [latex]10[/latex] inches?
[latex]\frac{5 \pi}{3} \approx 5.24[/latex] 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 [latex]r[/latex] is given by
[latex]A=\pi r^{2}[/latex]
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 pie in the previous example?
What is the area of a [latex]60°[/latex] wedge cut from a blueberry pie of diameter [latex]10[/latex] inches?
[latex]\frac{25 \pi}{6} \approx 13.09 \ \text{square inches}[/latex]
Look up the definitions of new terms in the Glossary.
Circle.
Circumference of a Circle.
Area of a Circle.
The Pythagorean theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical notation, it can be expressed as:
[ c^2 = a^2 + b^2 ]
where [latex]ax[/latex] and [latex]b[/latex] are the lengths of the two legs of the right triangle, and [latex]c[/latex] is the length of the hypotenuse. This theorem is named after the ancient Greek mathematician Pythagoras, who was the first to prove its validity.
A rational number is any number that can be expressed as a ratio of two integers, such as [latex]3/5[/latex].
An irrational number is a real number that cannot be expressed as a ratio of two integers, and its decimal expansion is non-repeating and non-terminating.
A repeating decimal is a decimal number that has a block of one or more digits that repeat infinitely after the decimal point, such as 0.333..., 0.666..., or 0.142857142857... (where the block 142857 repeats).
In geometry, a circle is a two-dimensional shape that consists of all points that are a fixed distance (called the radius) from a given point in a plane.
In geometry, the radius of a circle is the distance between the center of the circle and any point on its circumference. It is usually denoted by the letter “r”.
The unit circle is a circle with a radius of one centered at the origin [latex](0,0)[/latex] of a coordinate plane, often used in trigonometry and geometry as a reference circle. It has equations [latex]x^2 + y^2 = 1[/latex].
In geometry, the circumference of a circle is the distance around its outer perimeter or boundary. It can be calculated using the formula [latex]C = 2 \pi r[/latex]
In geometry, arc length is the length of a portion of the circumference of a circle, measured along the arc. It is denoted by “s” and can be calculated using the formula [latex]s = r theta[/latex] , where “r” is the radius of the circle and “θ” is the central angle (in radians) that subtends the arc.
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