# College Algebra

## Educators

### Problem 1

On the real number line the origin is assigned the number______.

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### Problem 2

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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### Problem 3

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

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### Problem 4

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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### Problem 5

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

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### Problem 6

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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### Problem 7

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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### Problem 8

The coordinate axes divide the $x y$ -plane into four sections called___.

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### Problem 9

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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### Problem 10

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

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### Problem 11

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

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### Problem 12

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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### Problem 13

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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### Problem 14

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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### Problem 15

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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### Problem 16

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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### Problem 17

Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.
CAN'T COPY THE FIGURE

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### Problem 18

Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.
CAN'T COPY THE FIGURE

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### Problem 19

Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.
CAN'T COPY THE FIGURE

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### Problem 20

Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$.
CAN'T COPY THE FIGURE

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### Problem 21

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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### Problem 22

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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### Problem 23

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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### Problem 24

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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### Problem 25

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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### Problem 26

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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### Problem 27

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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### Problem 28

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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### Problem 29

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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### Problem 30

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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### Problem 31

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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### Problem 32

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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### Problem 33

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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### Problem 34

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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### Problem 35

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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### Problem 36

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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### Problem 37

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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### Problem 38

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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### Problem 39

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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### Problem 40

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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### Problem 41

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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### Problem 42

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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### Problem 43

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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### Problem 44

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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### Problem 45

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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### Problem 46

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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### Problem 47

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

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### Problem 48

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

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### Problem 49

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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### Problem 50

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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### Problem 51

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 $C=(4,4)$.
CAN'T COPY THE FIGURE

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### Problem 52

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?
CAN'T COPY THE FIGURE

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### Problem 53

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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### Problem 54

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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### Problem 55

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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### Problem 56

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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### Problem 57

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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### Problem 58

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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### Problem 59

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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### Problem 60

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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### Problem 61

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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### Problem 62

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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### Problem 63

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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### Problem 64

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.
CAN'T COPY THE FIGURE

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### Problem 65

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.
CAN'T COPY THE FIGURE
(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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