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2) Examples are worked through, such as finding the coordinates of points, slopes of lines, and equations of lines and circles. Practice problems and their solutions are also provided.
3) Key topics covered include the Cartesian plane, distance between points, slope and equations of lines, parallel and perpendicular lines, conic sections including parabolas, circles, and ellipses, and their defining properties and equations.
Distinguish equations representing the circles and the conics; use the properties of a particular geometry to sketch the graph in using the rectangular or the polar coordinate system. Furthermore, to be able to write the equation and to solve application problems involving a particular geometry.
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Jr imp, Maths IB Important, Mathematics IB, Mathematics, Jr. Maths, Mathematics AP board, Mathematics important, Maths AP Board, Inter Maths IB, Inter Maths IB Important.
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MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...
Math - analytic geometry
1. I N F I N I T H I N K Page 1
Refresher Course
Content Area: MATHEMATICS
Focus: ANALYTIC GEOMETRY
Competencies:
1. Solve problems involving coordinates of a point, midpoint of a line segment, and distance between
two points.
2. Determine the equation of the line relative to given conditions: slope of a line given its graph, or its
equation, or any two points on it.
3. Determine the equation of a non-vertical line given a point on it and the slope of a line, which is
either parallel or perpendicular to it.
4. Solve problems involving
a. the midpoint of a line segment, distance between two points, slopes of lines, distance
between a point and a line, and segment division.
b. a circle, parabola, ellipse, and hyperbola.
5. Determine the equations and graphs of a circle, parabola, ellipse and hyperbola.
I. The Cartesian Plane
Below is a diagram of a Cartesian plane or a rectangular coordinate system, or a coordinate plane.
An ordered pair of real numbers, called the coordinates of a point, locates a point in the Cartesian plane.
Each ordered pair corresponds to exactly one point in the Cartesian plane.
The following are the points in the figure on the right:
A(-6,3), B(-2,-3), C (4,-2), D(3,4), E(0,5), F(-3,0).
For numbers 1-2, use the following condition: Two insects M and T are initially at a point A(-4, -7) on
a Cartesian plane.
1. If M traveled 7 units to the right and 8 units downward, at what point is it now?
Solution: (-4+7, -7-8) or (-3,-15)
2. If T traveled 5 units to the left and 11 units downward, at what point is it now?
Solution: (-4-5, -7-11) or (-9, -18)
II. The Straight Line
A. Distance Between Two Points
A. The distance between two points (x1,y1) and (x2,y2) is given by 2
21
2
21 )()( yyxx −+− .
Example: Given the points A(2,1) and B(5,4). Determine the length AB.
Solution: AB = ( ) ( ) ( ) ( ) 1899394152
2222
=+=−+−=−+− or 29• or 23 .
Exercises: For 1-2, use the following condition: Two insects L and O are initially at a point (-1,3) on a
Cartesian plane.
The two axes separate the plane into four
regions called quadrants. Points can lie in one of
the four quadrants or on an axis. The points on
the x-axis to the right of the origin correspond to
positive numbers; while to the left of the origin
correspond to negative numbers. The points on
the y-axis above the origin correspond to positive
numbers; while below the origin correspond to
negative numbers.
2. I N F I N I T H I N K Page 2
1. If L traveled 5 units to the left and 4 units upward, at what point is it now?
A) (-6, 7) B) (4, 7) C) (-6, -1) D) (4, -7)
2. If O traveled 6 units to the right and 2 units upward, at what point is it now?
A) (7, 5) B) (5,5) C) (-7, 5) D) (-5, -5)
3. Two buses leave the same station at 9:00 p.m. One bus travels at the rate of 30 kph and the other travels at 40
kph. If they go on the same direction, how many km apart are the buses at 10:00 p.m.?
A) 70 km B) 10 km C) 140 km D) 50 km
4. Two buses leave the same station at 8:00 a.m. One bus travels at the rate of 30 kph and the other travels at 40
kph. If they go on opposite direction, how many km apart are the buses at 9:00 a.m.?
A) 70 km B) 10 km C) 140 km D) 50 km
5. Two buses leave the same station at 7:00 a.m. One bus travels north at the rate of 30 kph and the other travels
east at 40 kph. How many km apart are the buses at 8:00 a.m.?
A) 70 km B) 10 km C) 140 km D) 50 km
6. Which of the following is true about the quadrilateral with vertices A(0,0), B(-2,1), C(3,4) and D(5,3)?
i) AD and BC are equal
ii) BD and AC are equal
iii) AB and CD are equal
A) both i and iii B) ii only C) both ii and iii D) i, ii, and iii
7. What is the distance between (-5,-8) and (10,0)?
A) 17 B) 13 C) 23 D) -0.5
B. Slope of a line
a) The slope of the non-vertical line containing A(x1,y1) and B(x2,y2) is
21
21
xx
yy
m
−
−
= or
12
12
xx
yy
m
−
−
= .
b) The slope of the line parallel to the x-axis is 0.
c) The slope of the line parallel to the y-axis is undefined.
d) The slope of the line that leans to the right is positive.
e) The slope of the line that leans to the left is negative.
C. The Equation of the line
In general, a line has an equation of the form ax + by + c = 0 where a, b, c are real numbers and that a
and b are not both zero.
D. Different forms of the Equation of the line
• General form: ax + by + c = 0.
• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
• Point slope form: )( 11 xxmyy −=− where (x1, y1) is any point on the line.
• Two point form: )( 1
12
12
1 xx
xx
yy
yy −
−
−
=− where (x1, y1) and (x2, y2) are any two points on the line.
• Intercept form: 1=+
b
y
a
x
where a is the x-intercept and b is the y-intercept.
Reminders:
• A line that leans to the right has positive slope. The steeper the line, the higher the slope is.
p q r
The slopes of lines p, q, r are all positive. Of the three slopes, the slope of line p is the lowest; the slope of r is
the highest.
3. I N F I N I T H I N K Page 3
• A line that leans to the left has negative slope. The steeper the line, the lower the slope is.
t s u
The slopes of lines t, s, and u are all negative. Of the three slopes, t is the highest, while u has the lowest.
Exercises
1. What is the slope of 5 4 12 0x y- + = ?
A)1.25 B) -1.25 C) 0.8 D) -0.8
2. What is the slope of x = -9?
A) 4 B) 1 C) 0 D) undefined
3. What is the slope of y= 12?
A) 7 B) 1 C) 0 D) undefined
4. What is the slope of 1
4 9
x y
+ = ?
A) 4.0 B) 2.25 C) - 4.0 D) - 2.25
E. Parallel and Perpendicular lines
Given two non-vertical lines p and q so that p has slope m1 and q has slope m2.
• If p and q are parallel, then m1 = m2.
• If p and q are perpendicular to each other, then m1m2 = -1.
F. Segment division
Given segment AB with A(x1,y1) and B(x2,y2).
• The midpoint M of segment AB is )
2
,
2
( 2121 yyxx
M
++
.
• If a point P divides AB in the ratio
2
1
r
r
so that
2
1
r
r
PB
AP
= , then the coordinates of P(x,y) can be obtained
using the formula
21
1221
rr
xrxr
x
+
+
= and
21
1221
rr
yryr
y
+
+
= .
G. Distance of a point from a line
The distance of a point A(x1,y1) from the line Ax + By + C = 0 is given by
22
11
BA
CByAx
d
+
++
= .
Exercises
1. Write an equation in standard form for the line passing through (–2,3) and (3,4).
a. 5x – y = -13 b. x – 5y = 19 c. x – y = -5 d. x – 5y = –17
2. Write an equation in slope intercept form for the line with a slope of 3 and a y-intercept of 28.
a. y = –3x + 28 b. y = 0.5x + 28 c. y = 3x + 28 d. y = 3x + 21
3. Write the equation in standard form for a line with slope of 3 and a y-intercept of 7.
a. 3x – y = –7 b. 3x + y = 7 c. 3x + y = 7 d. –3x + y = –7
4. Which of the following best describes the graphs of 2x – 3y = 9 and 6x – 9y = 18?
a. Parallel b. Perpendicular c. Coinciding d. Intersecting
5. Write the standard equation of the line parallel to the graph of x – 2y – 6 = 0 and passing through (0,1).
a. x + 2y = –2 b. 2x – y = –2 c. x – 2y = –2 d. 2x + y = –2
4. I N F I N I T H I N K Page 4
x
y
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
6. Write the equation of the line perpendicular to the graph of x = 3 and passing through (4, –1).
a. x – 4 = 0 b. y + 1 = 0 c. x + 1 = 0 d. y – 4 = 0
7. For what value of d will the graph of 6x + dy = 6 be perpendicular to the graph 2x – 6y = 12?
a. 0.5 b. 2 c. 4 d. 5
III. Conic Section
A conic section or simply conic, is defined as the graph of a second-degree equation in x and y.
In terms of locus of points, a conic is defined as the path of a point, which moves so that its distance
from a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus of the
conic, the fixed line is called the directrix of the conic, and the constant ratio is called the eccentricity, usually
denoted by e.
If e < 1, the conic is an ellipse. (Note that a circle has e=0.)
If e = 1, the conic is a parabola.
If e > 1, the conic is hyperbola.
A. The Circle
1. A circle is the set of all points on a plane that are equidistant from a fixed point on the plane. The fixed
point is called the center, and the distance from the center to any point of the circle is called the radius.
2. Equation of a circle
a) general form: x2
+ y2
+ Dx + Ey + F = 0
b) center-radius form: (x – h)2
+ (y – k)2
= r2
where the center is at (h,k) and the radius is equal to r.
3. Line tangent to a circle
A line tangent to a circle touches the circle at exactly one point called the point of tangency. The tangent
line is perpendicular to the radius of the circle, at the point of tangency.
Exercises
For items 1-2, use the illustration on the right.
1. Which of the following does NOT lie on the circle?
a. (3,-1) b. (3,0)
c. (2,-1) d. (3,-2)
2. What is the equation of the graph?
a. ( ) 132
=−+ xy b. ( ) 13)1( 2
=−+− xy
c. ( ) 13)1( 2
=−++ xy d. ( ) 13)1( 2
=+++ xy
B. The Parabola
1. Definition. A parabola is the set of all points on a plane that are equidistant from a
fixed point and a fixed line of the plane. The fixed point is called the focus and the fixed line is the
directrix.
2. Equation and Graph of a Parabola
a) The equation of a parabola with vertex at the origin and focus at (a,0) is y2
= 4ax. The parabola
opens to the right if a > 0 and opens to the left if a < 0.
b) The equation of a parabola with vertex at the origin and focus at (0,a) is x2
= 4ay. The parabola
opens upward if a > 0 and opens downward if a < 0.
c) The equation of a parabola with vertex at (h , k) and focus at (h + a, k) is (y – k)2
= 4a(x – h).
The parabola opens to the right if a > 0 and opens to the left if a < 0.
d) The equation of a parabola with vertex at (h , k) and focus at (h, k + a) is (x – h)2
= 4a(y – k).
5. I N F I N I T H I N K Page 5
e) The parabola opens upward if a > 0 and opens downward if a < 0.
f) Standard form: (y – k)2
= 4a(x – h) or (x – h)2
= 4a(y – k)
g) General form: y2
+ Dx + Ey + F = 0, or x2
+ Dx + Ey + F = 0
3. Parts of a Parabola
a) The vertex is the point, midway between the focus and the directrix.
b) The axis of the parabola is the line containing the focus and perpendicular to the directrix. The
parabola is symmetric with respect to its axis.
c) The latus rectum is the chord drawn through the focus and parallel to the directrix (and therefore
perpendicular to the axis) of the parabola.
d) In the parabola y2
=4ax, the length of latus rectum is 4a, and the endpoints of the latus rectum are (a,
-2a) and (a, 2a).
In the figure at the right, the vertex of the parabola is the origin,
the focus is F(a,o), the directrix is the line containing 'LL ,
the axis is the x-axis, the latus rectum is the line containing 'CC .
The graph of yx
3
162
−= . The graph of (y-2)2
= 8 (x-3).
C. Ellipse
1. An ellipse is the set of all points P on a plane such that the sum of the distances of P from two fixed points
F’ and F on the plane is constant. Each fixed point is called focus (plural: foci).
2. Equation of an Ellipse
a) If the center is at the origin, the vertices are at (± a, 0), the foci are at (± c,0), the endpoints of the
minor axis are at (0, ± b) and 222
cab −= , then the equation is 12
2
2
2
=+
b
y
a
x
.
b) If the center is at the origin, the vertices are at (0, ± a), the foci are at (0, ± c), the endpoints of the
minor axis are at (± b, 0) and 222
cab −= , then the equation is 12
2
2
2
=+
a
y
b
x
.
6. I N F I N I T H I N K Page 6
c) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is horizontal and
222
cab −= , then the equation is 1
)()(
2
2
2
2
=
−
+
−
b
ky
a
hx
.
d) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is vertical and
222
cab −= , then the equation is 1
)()(
2
2
2
2
=
−
+
−
b
hx
a
ky
.
3. Parts of an Ellipse
For the terms described below, refer to the ellipse
shown with center at O, vertices at V’(-a,0) and V(a,0),
foci at F’(-c,0) and F(c,0), endpoints of the minor axis
at B’(0,-b) and B(0,b), endpoints of one latus rectum
at G’ (-c,
a
b2
− ) and G(-c,
a
b2
) and the other at
H’ (c,
a
b2
− ) and G(c,
a
b2
).
a) The center of an ellipse is the midpoint of the segment joining the two foci. It is the intersection of
the axes of the ellipse. In the figure above, point O is the center.
b) The principal axis of the ellipse is the line containing the foci and intersecting the ellipse at its
vertices. The major axis is a segment of the principal axis whose endpoints are the vertices of the
ellipse. In the figure, VV ' is the major axis and has length of 2a units.
c) The minor axis is the perpendicular bisector of the major axis and whose endpoints are both on the
ellipse. In the figure, BB' is the minor axis and has length 2b units.
d) The latus rectum is the chord through a focus and perpendicular to the major axis. GG' and HH'
are the latus rectum, each with a length of
a
b2
2
.
The graph of 1
925
22
=+
yx
. The graph of 1
25
)1(
100
)2( 22
=
−
+
− yx
.
4. Kinds of Ellipses
a) Horizontal ellipse. An ellipse is horizontal if its principal axis is horizontal. The graphs above are both
horizontal ellipses.
b) Vertical ellipse. An ellipse is vertical if its principal axis is vertical.
D. The Hyperbola
1. A hyperbola is the set of points on a plane such that the difference of the distances of each point on the set
from two fixed points on the plane is constant. Each of the fixed points is called focus.
x
y
O
B(0,b
B’(0,-
F’(- F(c,0)V’(- V(a,0
),(
2
a
bc
),(
2
a
bc −
),(
2
a
bc−
),(
2
a
bc −−
x
y
O
(-4,0) (4,0) (5,0)
)
5
9(4,
(-5,0)
)
5
9(4,-
)
5
9(-4,
)
5
9(-4,-
(0, -3)
(0, 3)
x
y
O
(12,1)
(2,-4)
(-8,1)
(2,6)
(-6,4)
(2,1)
(8,5)
(8,3)
7. I N F I N I T H I N K Page 7
2. Equation of a hyperbola
a) If the center is at the origin, the vertices are at (± a, 0), the foci are at (± c,0), the endpoints of the minor
axis are at (0, ± b) and 222
acb −= , then the equation is 12
2
2
2
=−
b
y
a
x
.
b) If the center is at the origin, the vertices are at (0, ± a), the foci are at (0, ± c), the endpoints of the minor
axis are at (± b, 0) and 222
acb −= , then the equation is 12
2
2
2
=−
b
x
a
y
.
c) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is horizontal and
222
acb −= , then the equation is 1
)()(
2
2
2
2
=
−
−
−
b
ky
a
hx
.
d) If the center is at (h, k), the distance between the vertices is 2a, the principal axis is vertical and
222
acb −= , then the equation is 1
)()(
2
2
2
2
=
−
−
−
b
hx
a
ky
2. Parts of a hyperbola
For the terms described below, refer to the hyperbola shown which has its center at O, vertices at V’(-
a,0) and V(a,0), foci at F’(-c,0) and F(c,0) and endpoints of one latus rectum at G’ (-c,
a
b2
− ) and G(-c,
a
b2
) and the other at H’ (c,
a
b2
− ) and H(c,
a
b2
).
a) The hyperbola consists of two separate parts called branches.
b) The two fixed points are called foci. In the figure, the foci are at (± c,0).
c) The line containing the two foci is called the principal axis. In the
figure, the principal axis is the x-axis.
d) The vertices of a hyperbola are the points of intersection of the
hyperbola and the principal axis. In the figure, the vertices are at (± a,0).
e) The segment whose endpoints are the vertices is called the transverse axis. In the figure VV ' is the
transverse axis.
f) The line segment with endpoints (0,b) and (0,-b) where 222
acb −= is called the conjugate axis, and is a
perpendicular bisector of the transverse axis.
g) The intersection of the two axes is the center of the hyperbola .
h) The chord through a focus and perpendicular to the transverse axis is called a latus rectum. In the figure,
GG' is a latus rectum whose endpoints are G’ (-c,
a
b2
− ) and G(-c,
a
b2
) and has a length of
a
b2
2
.
3. The Asymptotes of a Hyperbola
Shown in the figure on the right is a hyperbola
with two lines as extended diagonals of the
rectangle shown.
8. I N F I N I T H I N K Page 8
These two diagonal lines are said to be the asymptotes of the curve, and are helpful in sketching the
graph of a hyperbola. The equations of the asymptotes associated with 12
2
2
2
=−
b
y
a
x
are x
a
b
y = and
x
a
b
y −= . Similarly, the equations of the asymptotes associated with 12
2
2
2
=−
b
x
a
y
are x
b
a
y = and x
b
a
y −= .
The graph of 1
279
22
=−
yx
. The graph of 1
279
22
=−
xy
.
PRACTICE EXERCISES
Directions: Choose the best answer from the choices given and write the corresponding letter of your choice.
For items 1-5, use the illustration on the right.
1. Which of the following are the coordinates of A?
a. (1,2) b. (2,1) c. (-3,3) d. (2,-3)
2. What is the distance between points M and T?
a. 61 units b. 6 sq. units c. 51 units d. 8 units
3. Which of the following points has the coordinates (-3,-1)
a. M b. A c. T d. H
4. Which of the following is the area of the triangle formed with vertices M, A and H?
a. 5 sq. units b. 10 sq. units c. 5 units d. 10 units
5. Which of the following is the equation of the line containing points A and T?
a. y= 2 b. x=2 c. y+2x=3 d. y-2x+3=0
6. Suppose that an isosceles trapezoid is placed on the Cartesian plane as shown
On the right, which of the following should be the coordinates of vertex V?
a. (a,b) b. (b+a, 0) c. (b-a,b) d. (b+a,b)
7. The points (-11,3), (3,8) and (-8,-2) are vertices of what triangle?
a. Isosceles b. Scalene c. Equilateral d. Right
x
y
F’(-6,0) O(-3,0) (3,0) F(6,0)
(6,9)
(6,-9)
x
y
F’(0,-6)
O
(0,3)
F(0,6)
(0,-3)
(-9,6) (9,6)
03 =− xy
03 =+ xy
x
1
2
3
1 2 3-3 -2 -1
-3
-2
-1
M
A
T
H
y
x
D(a,0) E(b,0)
O(0,b) V
0
0
9. I N F I N I T H I N K Page 9
x
y
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
8. What is the area of the triangle in #7?
a. 40.5 sq units b. 41.8 sq units c. 42 sq units d. 46.8 sq units
9. Which of the following sets of points lie on a straight line?
a. (2,3), (-4,7), (5,8) b. (-2,1), (3,2), (6,3) c. (-1,-4), (2,5), (7,-2) d. (4,1), (5,-2), (6,-5)
10. If the point (9,2) divides the segment of the line from P1(6,8) to P2(x2,y2) in the ratio r =
7
3
, give the
coordinates of P2.
a. (16,–12) b. (–16, 15) c. (14,15) d. (12,–12)
11. Give the fourth vertex, at the third quadrant, of the parallelogram whose three vertices are (-1,-5), (2,1) and
(1,5).
a. (-3,-2) c. (-3,-4) c. (-4,-1) d. (-2,-1)
12. The line segment joining A(-2,-1) and B(3,3) is extended to C. If BC = 3AB, give the coordinates of C.
a. (17,12) b. (15,17) c. (18,15) d. (12,18)
13. The line L2 makes an angle of 600
with the L1. If the slope of L1 is 1, give the slope of L2.
a. (3 + 20.5
) b. (2 + 20.5
) c. –(2 + 30.5
) d. –(3 + 30.5
)
14. The angle from the line through (-4,5) and (3,m) to the line (-2,4) and (9,1) is 1350
. Give the value of m.
a.7 b. 8 c. 9 d. 10
15. Which equation represents a line perpendicular to the graph of 2x + y = 2?
a. y = -0.5x – 2 b. y = –2x + 2 c. y = 2x – 2 d. y = 0.5x + 2
16. Which of the following is the y – intercept of the graph 2x – 2y + 8 = 0?
a. -4 b. -2 c. 2 d. 4
17. Which of the following may be a graph of x – y = a where a is a positive real number?
a. b. c. d.
18. Write an equation in standard form for a line with a slope of –1 passing through (2,1).
a. x + y = –3 b. –x + y = 3 c. x + y = 3 d. x – y = –3
For items 19-22, use the illustration on the right.
19. Which of the following are the coordinates of A?
a. (1,1) b. (1,-1)
c. (-1,1) d. (-1,-1)
20. What is the distance between points A and H?
a. 61 units b. 6 sq. units
c. 51 units d. 8 units
21. Which of the following points has the coordinates (-2,-2)?
a. M b. A
c. T d. H
22. Which of the following is the equation of the given graph?
a. ( ) 22
+= xy . b. ( ) 22
+−= xy . c. ( ) 22
−= xy . d. ( ) 22
−−= xy .
23. Which of the following is the equation of the line containing points M and T?
a. y= 2 b. x=2 c. y-2x-2=0 d. y+2x+2=0
24. What is the shortest distance of yx 82
= from 3=x ?
a. 1 unit b. 2 units c. 3 units d. 8 units
25. Which of the following is a focus of 1
412
22
=−
xy
?
y y y y
x x x x
A
HM
T
10. I N F I N I T H I N K Page 10
x
y
-30 -20 -10 0 10 20 30
-10
0
10
x
y
-30 -20 -10 0 10 20 30
-10
0
10
x
y
-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30
-15
-10
-5
0
5
10
15
x
y
-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30
-15
-10
-5
0
5
10
15
x
y
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
a. (0,-4) b. (-4,0) c. (0,4) d. (4,0)
26. What are the x-intercepts of 1
94
22
=+
yx
?
a. none b. 2± c. 3± d. 4±
27. Which of the following is a graph of a hyperbola?
a. b.
c. d.
28. Which of the following is an equation of an ellipse that has 10 as length of the major axis and has foci which
are 4 units away from the center?
a. 1
925
22
=+
xy
b. 1
169
22
=+
xy
c. 1
35
22
=+
xy
d. 1
2516
22
=+
xy
For items 29-31, consider the graph on the right.
29. Which of the following is the equation of
the graph?
a. 250025100 22
=+ xy
b. 250025100 22
=+ yx
c. 250025100 22
=− xy
d. 250025100 22
=− yx
30. What are the x-intercepts of the graph?
a. none b. 2±
c. 5± d. 10±
31. What kind of figure is shown on the graph?
a. circle b. ellipse c. hyperbola d. Parabola
32. Which of the following is the center of the graph
shown on the right?
a. (0,0) b. (0,10)
c. (10,0) c. (0,-10)
33. Which of the following is a focus of the graph
shown on the right?
a. (0,0) b. (0,10)
c. (0,5) c. (0,-10)
34. What is the area of the shaded region?
a. 4 units b. 4 square units
c. 16 units d. 16 square units
x
y
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2