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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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 _____

of $P$ and $y$____is the of $P$.

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The coordinate axes divide 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 by averaging the $x$ -coordinates and averaging the $y$ -coordinates of the endpoints.

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Plot each point in the xy-plane. Tell in 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) $F=(6,-3)$

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Plot each point in the xy-plane. Tell in 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 $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.

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Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.

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

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Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.

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

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Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.

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

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Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.

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

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Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.

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

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Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.

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

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Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.

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

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Find the distance $d\left(P_{1}, P_{2}\right)$ 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 $A B C$. Verify that the triangle is a right triangle. Find is area.

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

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

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

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

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

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

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

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

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

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Plot each point and form the triangle $A B C$. Verify that the triangle is a right triangle. Find is 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 $P_{2}$.

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

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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 $P_{2}$.

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

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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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Find all points on the $x$ -axis that are 6 units from the point $(4,-3)$

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Find all points on the $y$ -axis that are 6 units from the point $(4,-3)$

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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 is one in which all three sides are 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 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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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 (refer to Problem 52 ).

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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)?

CAN'T COPY THE FIGURE

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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 59. 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 $60 .$ Overlay a rectangular coordinate system on a Little 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 $(180,20),$ 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 Dodge Neon and a Mack truck leave an intersection at the same time. The Neon heads east at an average speed of 30 miles per hour, while the truck heads south at an average speed of 40 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.

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(a) Find an estimate for the desired intersection point.

(b) Find the length of the median for the midpoint found in part (a). See Problem 51.

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The figure illustrates how net sales of Wal-Mart Stores, Inc., have grown from 2002 through 2008 . Use the midpoint formula to estimate the net sales of Wal-Mart Stores, Inc., in $2005 .$ How does your result compare to the reported value of $\$ 282$ billion?

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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 1998 , the poverty threshold for a family of four with two children under the age of 18 years was $\$ 16,530 .$ In $2008,$ the poverty threshold for a family of four with two children under the age of 18 years was $\$ 21,834$. Assuming poverty thresholds increase in a straight-line fashion, use the midpoint formula to estimate the poverty threshold of a family of four with two children under the age of 18 in $2003 .$ How does your result compare to the actual poverty threshold in 2003 of $\$ 18,660 ?$

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