What fundamental theorem is used as the basis for deriving the Distance Formula?
Write the equation for a circle that has its center at the origin and a radius of \(r\).
What is a "unit circle"?
What is the "standard form" of the equation for a circle with center \((h, k)\) and radius \(r\)?
In which quadrant are the x-coordinates negative and the y-coordinates positive?
What two conditions define an angle in "standard position"?
Where is the terminal side of a 90° angle in standard position?
What is a "quadrantal angle"?
What does it mean for two angles to be "coterminal"?
Describe the direction of rotation (clockwise or counterclockwise) for a negative angle.
If you are given an angle of 50°, how would you find a negative angle that is coterminal with it?
Skills Refresher
Review the following skills you will need for this section.
Skills Refresher
For the following exercises, find the slope and \( y \)-intercept of the graph for the given equation (do not graph).
\( y = 5.2x - 1.9 \)
\( y = x \)
\( y = -8 \)
\( x = 7 \)
\( 12y - 10x = 5 \)
For the following exercises, write the equation of the line that satisfies the given conditions.
The line goes through the points \( \left( -2,6 \right) \) and \( \left( 3,-4 \right) \).
The line goes through the points \( \left( 5,-7 \right) \) and \( \left( 5,-5 \right) \).
For the following exercises, determine if the statement is true or false.
\(\sqrt{a^2+b^2}=a+b\)
\(\sqrt{36+64}=6+8\)
\(\sqrt{16 x^4}=4 x^2\)
\(\sqrt{2 x} \sqrt{3 y}=\sqrt{6 x y}\)
\(\sqrt{5 x}+\sqrt{3 x}=\sqrt{8 x}\)
\(\sqrt{4+N}=2+\sqrt{N}\)
\(\sqrt{\dfrac{x}{4}}=\dfrac{\sqrt{x}}{2}\)
\(\sqrt{\dfrac{3}{2}}=\dfrac{\sqrt{6}}{2}\)
For the following exercises, solve the equation. If no solution exists, state why.
\(x^2 - 81 = 0\)
\(2 y^2 - 98 = 0\)
\(x^2 + 12 = 8\)
\(-3x^2 + 19 = 10\)
Answers
\( m = 5.2 \quad \left( 0,-1.9 \right)\)
\( m = 1 \quad \left( 0,0 \right) \)
\( m = 0 \quad \left( 0,-8 \right) \)
The slope is undefined and there is not a \( y \)-intercept.
\( m = \frac{5}{6} \quad \left( 0,\frac{5}{12} \right) \)
\( y = -2x + 2 \)
\( x = 5 \)
False
False
True
True
False
False
True
True
\( x = \pm 9 \)
\( y = \pm 7 \)
No solution because \( x^2 \) should never be negative, but when solving, we get \( x^2 = -4 \).
\( x = \pm \sqrt{3} \)
Homework
Concept Check
Explain why the Distance Formula,\[d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2},\nonumber \]cannot be simplified to\[(x_2-x_1)+(y_2-y_1).\nonumber \]
What is a unit circle, and what is its equation?
In which quadrant are the \( x \)-values for points negative but the \( y \)-values are positive?
Explain what it means for angles to be coterminal. Demonstrate by creating three coterminal angles, one of which is a negative angle.
True or False? For the following exercises, determine if the statement is true or false. If true, cite the definition or theorem stated in the text supporting your claim. If false, explain why it is false and, if possible, correct the statement.
Positive angles open clockwise.
Basic Skills
For the following exercises, find the distance between the points. Give your answer as an exact value, then as a decimal rounded to hundredths.
(MyOpenMath) Write an expression describing all the angles that are coterminal with \( 252^{ \circ } \).
Synthesis Questions
Show that the triangle with vertices \((0, 0)\), \((6, 0)\), and \((3, 3)\) is an isosceles right triangle, that is, a right triangle with two sides of the same length.
Two opposite vertices of a square are \(A(−9, −5)\) and \(C(3, 3)\).
Find the length of a diagonal of the square.
Find the length of the side of the square.
Find all of the points on the line \(\ y = 2x + 1\) which are \(4\) units from the point \((-1, 3)\).
Sketch a triangle with vertices \((10, 1)\), \((3, 1)\), and \((5, 9)\), and find its perimeter. Round your answer to tenths.
Sketch a triangle with vertices \((−1, 5)\), \((8, −7)\), and \((4, 1)\), and find its perimeter. Round your answer to tenths.
For the following exercises, interpret the equations as statements about distance.
\(\sqrt{(x-4)^2+(y+1)^2}=3\)
\(\sqrt{(-2-h)^2+(5-k)^2}=l\)
For the following exercises, use the graph to answer the question.
For the following exercises, graph the equation and find the circumference of the circle.
\(x^2+y^2 = 36\)
\(x^2+y^2=16\)
\(4 x^2+4 y^2=16\)
Give the coordinates of two points on the circle in the previous exercise that have \(y = −4\). Plot those points on your graph.
\(2 x^2+2 y^2=18\)
Give the coordinates of two points on the circle in the previous exercise that have \(x = −2\). Plot those points on your graph.
Applications
Distance. Paige is sailing and is currently 3 miles west and 5 miles south of the harbor. She heads directly towards an island 8 miles west and 7 miles north of the harbor. How far is Paige from the island?
Distance. Nam is 100 meters east and 250 meters north of Kristen. He walks directly towards a tree 220 meters east and 90 meters north of Kristen. How far is Nam from the tree?
Through what angle does the hour hand of a clock rotate between 2 pm and 10 pm?
Through what angle does the hour hand of a clock rotate between 2 am and 10 pm?
Computing Angles.
Through what angle does the hour hand of a clock rotate between 3:25 am and 3:50 am?
Through what angle does the hour hand of a clock rotate between 4:10 pm and 6:25 pm?
Astronomy. The Distance Formula can be modified to compute distances between objects in three-dimensional space. To do so, we must expand the Cartesian coordinate system to include one more spatial dimension. In three-space, coordinates are listed in ordered triples, such as \( (x,y,z) \). If we have two points, \( P(x_1,y_1,z_1) \) and \( Q(x_2,y_2,z_2) \), in three-dimensional space, the distance between these points is\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}. \nonumber \]Suppose our Sun is given the very special coordinate of \( (0,0,0) \) (thereby affirming that we are the center of the universe). Compute the distance between each star (these computed distances are in light-years). Round answers to the nearest hundredth of a light-year.
Computing Angles. For the following exercises, calculate the degree measure of the unknown angle and sketch the angle in standard position.
Challenge Problems
In this problem, we’ll show that any angle inscribed in a semi-circle must be a right angle. The figure shows a triangle inscribed in a unit circle, one side lying on the diameter of the circle and the opposite vertex at point \((p, q)\) on the circle.
What are the coordinates of the other two vertices of the triangle? What is the length of the side joining those vertices?
Use the Distance Formula to compute the lengths of the other two sides of the triangle.
Show that the sides of the triangle satisfy the Pythagorean Theorem, \(a^2 + b^2 = c^2\).
We shall now prove that \(\ y=m_{1} x+b_{1}\) is perpendicular to \(\ y=m_{2} x+b_{2}\) if and only if \(\ m_{1} \cdot m_{2}=-1\). To make our lives easier we shall assume that \(\ m_{1}>0\) and \(\ m_{2}<0\). We can also "move" the lines so that their point of intersection is the origin without messing things up, so we’ll assume \(\ b_{1}=b_{2}=0\). (Take a moment with your classmates to discuss why this is okay.) Graphing the lines and plotting the points \(\ O(0, 0)\), \(\ P\left(1, m_{1}\right)\) and \(\ Q\left(1, m_{2}\right)\) gives us the following set up.
The line \(\ y=m_{1} x\) will be perpendicular to the line \(\ y=m_{2} x\) if and only if \(\ \triangle O P Q\) is a right triangle. Let \(\ d_{1}\) be the distance from \(\ O\) to \(\ P\), let \(\ d_{2}\) be the distance from \(\ O\) to \(\ Q\) and let \(\ d_{3}\) be the distance from \(\ P\) to \(\ Q\). Use the Pythagorean Theorem to show that \(\ \triangle O P Q\) is a right triangle if and only if \(\ m_{1} \cdot m_{2}=-1\) by showing \(\ d_{1}^{2}+d_{2}^{2}=d_{3}^{2}\) if and only if \(\ m_{1} \cdot m_{2}=-1\).
