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Section 1
The Distance and Midpoint Formulas
On the real number line, the origin is assigned the number _____.
If -3 and 5 are the coordinates of two points on the real number line, the distance between these points is ____
If 3 and 4 are the legs of a right triangle, the hypotenuse is ____.
Use the converse of the Pythagorean Theorem to show that a triangle whose sides are of lengths $11,60,$ and 61 is a right triangle.
The area $A$ of a triangle whose base is $b$ and whose altitude is $h$ is $A=$ _____.
Two triangles are congruent if two angles and the included side of one equals two angles and the included side of the other.
If $(x, y)$ are the coordinates of a point $P$ in the $x y$ -plane, then $x$ is called the _______ of $P$, and $y$ is the ___________of $P$.
The coordinate axes partition the $x y$ -plane into four sections called ______.
If three distinct points $P, Q,$ and $R$ all lie on a line, and if $d(P, Q)=d(Q, R),$ then $Q$ is called the _____ of the line segment from $P$ to $R$.
The distance between two points is sometimes a negative number.
The point (-1,4) lies in quadrant IV of the Cartesian plane.
The midpoint of a line segment is found by averaging the $x$ -coordinates and averaging the $y$ -coordinates of the endpoints.
Which of the following statements is true for a point $(x, y)$ that lies in quadrant III?(a) Both $x$ and $y$ are positive.(b) Both $x$ and $y$ are negative.(c) $x$ is positive, and $y$ is negative.(d) $x$ is negative, and $y$ is positive.
Choose the expression that equals the distance between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$(a) $\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$(b) $\sqrt{\left(x_{2}+x_{1}\right)^{2}-\left(y_{2}+y_{1}\right)^{2}}$(c) $\sqrt{\left(x_{2}-x_{1}\right)^{2}-\left(y_{2}-y_{1}\right)^{2}}$(d) $\sqrt{\left(x_{2}+x_{1}\right)^{2}+\left(y_{2}+y_{1}\right)^{2}}$
Plot each point in the xy-plane. State which quadrant or on what coordinate axis each point lies.(a) $A=(-3,2)$(d) $D=(6,5)$(b) $B=(6,0)$(e) $E=(0,-3)$(c) $C=(-2,-2)$(f) $F=(6,-3)$
Plot each point in the xy-plane. State which quadrant or on what coordinate axis each point lies.(a) $A=(1,4)$(d) $D=(4,1)$(b) $B=(-3,-4)$(e) $E=(0,1)$(c) $C=(-3,4)$(f) $F=(-3,0)$
Plot each point in the xy-plane. State which quadrant or on what coordinate axis each point lies.Plot the points $(2,0),(2,-3),(2,4),(2,1),$ and $(2,-1) .$ Describe the set of all points of the form $(2, y),$ where $y$ is a real number.
Plot each point in the xy-plane. State which quadrant or on what coordinate axis each point lies.Plot the points $(0,3),(1,3),(-2,3),(5,3),$ and $(-4,3) .$ Describe the set of all points of the form $(x, 3),$ where $x$ is a real number.
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(3,-4) ; \quad P_{2}=(5,4)$$
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(-1,0) ; \quad P_{2}=(2,4)$$
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(-7,3) ; \quad P_{2}=(4,0)$$
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(2,-3) ; \quad P_{2}=(4,2)$$
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(5,-2) ; \quad P_{2}=(6,1)$$
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(-4,-3) ; \quad P_{2}=(6,2)$$
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(-0.2,0.3) ; \quad P_{2}=(2.3,1.1)$$
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(1.2,2.3) ; \quad P_{2}=(-0.3,1.1)$$
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(a, b) ; \quad P_{2}=(0,0)$$
Find the distance $d$ between the points $P_{1}$ and $P_{2}$.$$P_{1}=(a, a) ; \quad P_{2}=(0,0)$$
Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.$$A=(-2,5) ; \quad B=(1,3) ; \quad C=(-1,0)$$
Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.$$A=(-2,5) ; \quad B=(12,3) ; \quad C=(10,-11)$$
Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.$$A=(-5,3) ; \quad B=(6,0) ; \quad C=(5,5)$$
Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.$$A=(-6,3) ; \quad B=(3,-5) ; \quad C=(-1,5)$$
Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.$$A=(4,-3) ; \quad B=(0,-3) ; \quad C=(4,2)$$
Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.$$A=(4,-3) ; \quad B=(4,1) ; \quad C=(2,1)$$
Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.$$P_{1}=(3,-4) ; \quad P_{2}=(5,4)$$
Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.$$P_{1}=(-2,0) ; \quad P_{2}=(2,4)$$
Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.$$P_{1}=(-1,4) ; \quad P_{2}=(8,0)$$
Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.$$P_{1}=(2,-3) ; \quad P_{2}=(4,2)$$
Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.$$P_{1}=(7,-5) ; \quad P_{2}=(9,1)$$
Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.$$P_{1}=(-4,-3) ; \quad P_{2}=(2,2)$$
Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.$$P_{1}=(a, b) ; \quad P_{2}=(0,0)$$
Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.$$P_{1}=(a, a) ; \quad P_{2}=(0,0)$$
If the point (2,5) is shifted 3 units to the right and 2 units down, what are its new coordinates?
If the point (-1,6) is shifted 2 units to the left and 4 units up, what are its new coordinates?
Find all points having an $x$ -coordinate of 3 whose distance from the point (-2,-1) is $13 .$(a) By using the Pythagorean Theorem.(b) By using the distance formula.
Find all points having a $y$ -coordinate of -6 whose distance from the point (1,2) is 17(a) By using the Pythagorean Theorem.(b) By using the distance formula.
Find all points on the $x$ -axis that are 6 units from the point (4,-3)
Find all points on the $y$ -axis that are 6 units from the point (4,-3)
Suppose that $A=(2,5)$ are the coordinates of a point in the $x y$ -plane.(a) Find the coordinates of the point if $A$ is shifted 3 units to the left and 4 units down.(b) Find the coordinates of the point if $A$ is shifted 2 units to the left and 8 units up.
Plot the points $A=(-1,8)$ and $M=(2,3)$ in the $x y$ -plane. If $M$ is the midpoint of a line segment $A B,$ find the coordinates of $B$.
The midpoint of the line segment from $P_{1}$ to $P_{2}$ is (-1,4) If $P_{1}=(-3,6),$ what is $P_{2} ?$
The midpoint of the line segment from $P_{1}$ to $P_{2}$ is (5,-4) . If $P_{2}=(7,-2),$ what is $P_{1} ?$
Geometry The medians of a triangle are the line segments from each vertex to the midpoint of the opposite side (see the figure). Find the lengths of the medians of the triangle with vertices at $A=(0,0), B=(6,0),$ and $C=(4,4)$
Geometry An equilateral triangle has three sides of equal length. If two vertices of an equilateral triangle are (0,4) and (0,0) find the third vertex. How many of these triangles are possible?
Find the length of each side of the triangle determined by the three points $P_{1}, P_{2},$ and $P_{3}$. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length). $$P_{1}=(2,1) ; \quad P_{2}=(-4,1) ; \quad P_{3}=(-4,-3)$$
Find the length of each side of the triangle determined by the three points $P_{1}, P_{2},$ and $P_{3}$. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length).$$P_{1}=(-1,4) ; \quad P_{2}=(6,2) ; \quad P_{3}=(4,-5)$$
Find the length of each side of the triangle determined by the three points $P_{1}, P_{2},$ and $P_{3}$. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length).$$P_{1}=(-2,-1) ; \quad P_{2}=(0,7) ; \quad P_{3}=(3,2)$$
Find the length of each side of the triangle determined by the three points $P_{1}, P_{2},$ and $P_{3}$. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length).$$P_{1}=(7,2) ; \quad P_{2}=(-4,0) ; \quad P_{3}=(4,6)$$
A major league baseball "diamond" is actually a square 90 feet on a side (see the figure). What is the distance directly from home plate to second base (the diagonal of the square)?
Refer to Problem 63. Overlay a rectangular coordinate system on a major league baseball diamond so that the origin is at home plate, the positive $x$ -axis lies in the direction from home plate to first base, and the positive $y$ -axis lies in the direction from home plate to third base.(a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.(b) If the right fielder is located at (310,15) how far is it from the right fielder to second base?(c) If the center fielder is located at $(300,300),$ how far is it from the center fielder to third base?
Refer to Problem 64. Overlay $\begin{array}{llllll}\text { a rectangular coordinate } & \text { system } & \text { on a } & \text { little } & \text { League }\end{array}$ baseball diamond so that the origin is at home plate, the positive $x$ -axis lies in the direction from home plate to first base, and the positive $y$ -axis lies in the direction from home plate to third base.(a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.(b) If the right fielder is located at ( 180,20 ), how far is it from the right fielder to second base?(c) If the center fielder is located at $(220,220),$ how far is it from the center fielder to third base?
A Ford Focus and a Freightliner Cascadia truck leave an intersection at the same time. The Focus heads east at an average speed of 60 miles per hour, while the Cascadia heads south at an average speed of 45 miles per hour. Find an expression for their distance apart $d$ (in miles) at the end of $t$ hours.
A hot-air balloon, headed due east at an average speed of 15 miles per hour and at a constant altitude of 100 feet, passes over an intersection (see the figure). Find an expression for the distance $d$ (measured in feet) from the balloon to the intersection $t$ seconds later.
The layout of a Little League playing field is a square 60 feet on a side. How far is it directly from home plate to second base (the diagonal of the square)?
When a draftsman draws three lines that are to intersect at one point, the lines may not intersect as intended and subsequently will form an error triangle. If this error triangle is long and thin, one estimate for the location of the desired point is the midpoint of the shortest side. The figure shows one such error triangle.(a) Find an estimate for the desired intersection point.(b) Find the distance from (1.4,1.3) to the midpoint found in part (a).
Net Sales The figure illustrates the net sales growth of costco Wholesale Corporation from 2013 through 2017 . Use the midpoint formula to estimate the net sales of costco Wholesale Corporation in 2015. How does your result compare to the reported value of $\$ 113.67$ billion?
Poverty thresholds are determined by the U.S. Census Bureau. A poverty threshold represents the minimum annual household income for a family not to be considered poor. In $2009,$ the poverty threshold for a family of four with two children under the age of 18 years was $\$ 21,756 .$ In $2017,$ the poverty threshold for a family of four with two children under the age of 18 years was $\$ 24,858$. Assuming that poverty thresholds increase in a straight-line fashion, use the midpoint formula to estimate the poverty threshold for a family of four with two children under the age of 18 in $2013 .$ How does your result compare to the actual poverty threshold in 2013 of $\$ 23,624 ?$
Verify that the points $(0,0),(a, 0)$ and $\left(\frac{a}{2}, \frac{\sqrt{3} a}{2}\right)$ are the vertices of an equilateral triangle.Then show that the midpoints of the three sides are the vertices of a second equilateral triangle.
Find the midpoint of each diagonal of a square with side of length $s$. Draw the conclusion that the diagonals of a square intersect at their midpoints.[Hint: Use $(0,0),(0, s),(s, 0),$ and $(s, s)$ as the vertices of the square. $]$
A point $P$ is equidistant from (-5,1) and $(4,-4) .$ Find the coordinates of $P$ if its $y$ -coordinate is twice its $x$ -coordinate.
Geometry For any parallelogram, prove that the sum of the squares of the lengths of the sides equals the sum of the squares of the lengths of the diagonals.
Write a paragraph that describes a Cartesian plane. Then write a second paragraph that describes how to plot points in the Cartesian plane. Your paragraphs should include the terms "coordinate axes," "ordered pair," "coordinates," "plot," " $x$ -coordinate," and " $y$ -coordinate."
Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Determine the domain of the variable $x$ in the expression:$$\frac{3 x+1}{2 x-5}$$
Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Find the value of $\frac{x^{2}-3 x y+2}{5 x-2 y}$ if $x=4$ and $y=7$
Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Express $(5 x-2)(3 x+7)$ as a polynomial in standard form.
Express $(5 x-2)(3 x+7)$ as a polynomial in standard form.
Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Reduce the rational expression to lowest terms:$$\frac{3 x^{2}+7 x+2}{3 x^{2}-11 x-4}$$
Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Factor the expression completely:$$6(x+1)^{3}(2 x-5)^{7}-5(x+1)^{4}(2 x-5)^{6}$$
Find the real solution $(s),$ if any, of each equation.Find the real solution $(s),$ if any, of each equation.$$3 x^{2}-7 x-20=0$$
Find the real solution $(s),$ if any, of each equation.$$\frac{x}{x+3}+\frac{1}{x-3}=1$$
Find the real solution $(s),$ if any, of each equation.$$|7 x-4|=31$$
Find the real solution $(s),$ if any, of each equation.Solve the inequality $5(x-3)+2 x \geq 6(2 x-3)-7$ Express the solution using interval notation. Graph the solution set.
Find the real solution $(s),$ if any, of each equation.Multiply $(7+3 i)(1-2 i) .$ Write the answer in the form $a+b i$.