On the real number line, the origin is assigned the number_______________.

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If -3 and 5 are the coordinates of two points on the real number line, the distance between these points is ____________.

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If 3 and 4 are the legs of a right triangle, the hypotenuse is ________________.

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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.

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The area $A$ of a triangle whose base is $b$ and whose altitude is $h$ is $A=$ ___________________.

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True or False Two triangles are congruent if two angles and the included side of one equals two angles and the included side of the other.

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If $(x, y)$ are the coordinates of a point $P$ in the $x y$ -plane, then $x$ is called the is the ______________________ of $P$, and $y$ is the _______________________ of $P$.

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The coordinate axes partition the $x y$ -plane into four sections called _______________.

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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 $R$.

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True or False. The distance between two points is sometimes a negative number.

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True or False The point ( - 1 , 4 ) lies in quadrant IV of the Cartesian plane.

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True or False. The midpoint of a line segment is found byaveraging the $x$ -coordinates and averaging the $y$ -coordinates of the endpoints.

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Multiple Choice. 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.

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Multiple Choice 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) $\sqrt{\left(x_{2}+x_{1}\right)^{2}+\left(y_{2}+y_{1}\right)^{2}}$

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Plot each point in the xy-plane. State which quadrant or on what coordinate axis each point lies.

(a) $A=(-3,2)$

(b) $B=(6,0)$

(c) $C=(-2,-2)$

(d) $D=(6,5)$

(e) $E=(0,-3)$

(f) $F=(6,-3)$

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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) $F=(-3,0)$

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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.

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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.

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(3,-4) ; \quad P_{2}=(5,4)$$

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(-1,0) ; \quad P_{2}=(2,4)$$

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(-7,3) ; \quad P_{2}=(4,0)$$

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(2,-3) ; \quad P_{2}=(4,2)$$

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(5,-2) ; \quad P_{2}=(6,1)$$

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(-4,-3) ; \quad P_{2}=(6,2)$$

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(-0.2,0.3) ; \quad P_{2}=(2.3,1.1)$$

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(1.2,2.3) ; \quad P_{2}=(-0.3,1.1)$$

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(a, b) ; \quad P_{2}=(0,0)$$

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Find the distance $d$ between the points $P_{1}$ and $P_{2}$.

$$P_{1}=(a, a) ; \quad P_{2}=(0,0)$$

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Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.

$$A=(-2,5) ; \quad B=(1,3) ; \quad C=(-1,0)$$

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Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.

$$A=(-2,5) ; \quad B=(12,3) ; \quad C=(10,-11)$$

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Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.

$$A=(-5,3) ; \quad B=(6,0) ; \quad C=(5,5)$$

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Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.

$$A=(-6,3) ; \quad B=(3,-5) ; \quad C=(-1,5)$$

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Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.

$A=(4,-3) ; \quad B=(0,-3) ; \quad C=(4,2)$

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Plot each point and form the triangle $A B C$. Show that the triangle is a right triangle. Find its area.

$A=(4,-3) ; \quad B=(4,1) ; \quad C=(2,1)$

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Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.

$P_{1}=(3,-4) ; \quad P_{2}=(5,4)$

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Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.

$P_{1}=(-2,0) ; \quad P_{2}=(2,4)$

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Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.

$P_{1}=(-1,4) ; \quad P_{2}=(8,0)$

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Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.

$P_{1}=(2,-3) ; \quad P_{2}=(4,2)$

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Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.

$P_{1}=(7,-5) ; \quad P_{2}=(9,1)$

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Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.

$P_{1}=(-4,-3) ; \quad P_{2}=(2,2)$

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Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.

$P_{1}=(a, b) ; \quad P_{2}=(0,0)$

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Find the midpoint of the line segment joining the points $P_{1}$ and $P_{2}$.

$P_{1}=(a, a) ; \quad P_{2}=(0,0)$

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If the point (2,5) is shifted 3 units to the right and 2 units down, what are its new coordinates?

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If the point (-1,6) is shifted 2 units to the left and 4 units up, what are its new coordinates?

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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.

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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.

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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.

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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 $B$.

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The midpoint of the line segment from $P_{1}$ to $P_{2}$ is (-1,4) . If $P_{1}=(-3,6),$ what is $P_{2} ?$

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The midpoint of the line segment from $P_{1}$ to $P_{2}$ is (5,-4) . If $P_{2}=(7,-2),$ what is $P_{1} ?$

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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 $C=(4,4)$

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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?

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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. )

$P_{1}=(2,1) ; \quad P_{2}=(-4,1) ; \quad P_{3}=(-4,-3)$

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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. )

$P_{1}=(-1,4) ; \quad P_{2}=(6,2) ; \quad P_{3}=(4,-5)$

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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. )

$P_{1}=(-2,-1) ; \quad P_{2}=(0,7) ; \quad P_{3}=(3,2)$

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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. )

$P_{1}=(7,2) ; \quad P_{2}=(-4,0) ; \quad P_{3}=(4,6)$

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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 $) ?$

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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)?

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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?

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Refer to Problem 64. Overlay $\begin{array}{llllll}\text { a rectangular } & \text { coordinate } & \text { system } & \text { on } & \text { a } & \text { Little } & \text { League }\end{array}$ 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?

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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.

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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.

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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).

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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? Source: Costco Wholesale Corporation 2017 Annual Report

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Poverty Threshold 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 $\$ 23,624 ?$

Source: U.S. Census Bureau

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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.

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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.]

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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.

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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.

[Hint: Use $(0,0),(a, 0),(a+b, c),$ and $(b, c)$ as the vertices of the parallelogram.

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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."

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Determine the domain of the variable $x$ in the expression:

$$

\frac{3 x+1}{2 x-5}

$$

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Reduce the rational expression to lowest terms:

$$

\frac{3 x^{2}+7 x+2}{3 x^{2}-11 x-4}

$$

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Factor the expression completely:

$$

6(x+1)^{3}(2 x-5)^{7}-5(x+1)^{4}(2 x-5)^{6}

$$

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Find the real solution $(s),$ if any, of each equation.

$3 x^{2}-7 x-20=0$

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Find the real solution $(s),$ if any, of each equation.

$\frac{x}{x+3}+\frac{1}{x-3}=1$

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Solve the inequality $5(x-3)+2 x \geq 6(2 x-3)-7$ Express the solution using interval notation. Graph the solution set.

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