Download M1 End-of-Module Assessment Task - One Math

NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M1
GEOMETRY
Name
Date
1. Each of the illustrations on the next page shows in black a plane figure consisting of the letters F, R, E, and
D evenly spaced and arranged in a row. In each illustration, an alteration of the black figure is shown in
gray. In some of the illustrations, the gray figure is obtained from the black figure by a geometric
transformation consisting of a single rotation. In others, this is not the case.
a. Which illustrations show a single rotation?
b. Some of the illustrations are not rotations or even a sequence of rigid transformations. Select one
such illustration, and use it to explain why it is not a sequence of rigid transformations.
Module 1:
Congruence, Proof, and Constructions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M1
GEOMETRY
Module 1:
Congruence, Proof, and Constructions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M1
GEOMETRY
̅̅̅̅ bisects ∠𝐴𝐶𝐵, 𝐴𝐵 = 𝐵𝐶, 𝑚∠𝐵𝐸𝐶 = 90°, and 𝑚∠𝐷𝐶𝐸 = 42°.
2. In the figure below, 𝐶𝐷
Find the measure of ∠𝐴.
Module 1:
Congruence, Proof, and Constructions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M1
GEOMETRY
̅̅̅̅ is the angle bisector of ∠𝐵𝐴𝐶. ̅̅̅̅̅̅
̅̅̅̅̅̅ are straight lines, and 𝐴𝐷
̅̅̅̅ ∥ 𝑃𝐶
̅̅̅̅ .
3. In the figure below, 𝐴𝐷
𝐵𝐴𝑃 and 𝐵𝐷𝐶
Prove that 𝐴𝑃 = 𝐴𝐶.
Module 1:
Congruence, Proof, and Constructions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M1
GEOMETRY
̅̅̅̅ ≅ 𝐷𝐸
̅̅̅̅ ≅ 𝐷𝐹
̅̅̅̅ , 𝐴𝐶
̅̅̅̅ , and ∠𝐴 ≅ ∠𝐷.
4. △ 𝐴𝐵𝐶 and △ 𝐷𝐸𝐹, in the figure below are such that 𝐴𝐵
a. Which criteria for triangle congruence (ASA, SAS, SSS) implies that △ 𝐴𝐵𝐶 ≅△ 𝐷𝐸𝐹?
b. Describe a sequence of rigid transformations that shows △ 𝐴𝐵𝐶 ≅△ 𝐷𝐸𝐹.
Module 1:
Congruence, Proof, and Constructions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M1
GEOMETRY
5.
a.
̅̅̅̅. List the steps of the construction.
Construct a square 𝐴𝐵𝐶𝐷 with side 𝐴𝐵
Module 1:
Congruence, Proof, and Constructions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M1
GEOMETRY
b.
Three rigid motions are to be performed on square 𝐴𝐵𝐶𝐷. The first rigid motion is the reflection
through ̅̅̅̅
𝐵𝐷. The second rigid motion is a 90° clockwise rotation around the center of the square.
Describe the third rigid motion that will ultimately map 𝐴𝐵𝐶𝐷 back to its original position. Label the
image of each rigid motion 𝐴, 𝐵, 𝐶, and 𝐷 in the blanks provided.
̅̅̅̅
Rigid Motion 1 Description: Reflection through 𝐵𝐷
Rigid Motion 2 Description: 90° clockwise rotation around the
center of the square.
Rigid Motion 3 Description:
Module 1:
Congruence, Proof, and Constructions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M1
GEOMETRY
̅̅̅̅ and 𝐶𝐷
̅̅̅̅, respectively.
6. Suppose that 𝐴𝐵𝐶𝐷 is a parallelogram and that 𝑀 and 𝑁 are the midpoints of 𝐴𝐵
Prove that 𝐴𝑀𝐶𝑁 is a parallelogram.
Module 1:
Congruence, Proof, and Constructions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M2
GEOMETRY
Name
Date
̅̅̅̅, prove that
1. In the figure below, rotate △ 𝐸𝐴𝐵 about 𝐸 by 180° to get △ 𝐸𝐴′ 𝐵′ . If ̅̅̅̅̅̅
𝐴′ 𝐵′ ∥ 𝐶𝐷
△ 𝐸𝐴𝐵 ~ △ 𝐸𝐷𝐶.
Module 2:
Similarity, Proof, and Trigonometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M2
GEOMETRY
2. Answer the following questions based on the diagram below.
a.
b.
Find the sine and cosine values of angles 𝑟 and 𝑠. Leave the answers as fractions.
sin 𝑟° =
sin 𝑠° =
cos 𝑟° =
cos 𝑠° =
tan 𝑟° =
tan 𝑠° =
Why is the sine of an acute angle the same value as the cosine of its complement?
Module 2:
Similarity, Proof, and Trigonometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M2
GEOMETRY
c.
Determine the measures of the angles to the nearest tenth of a degree in the right triangles below.
i.
Determine the measure of ∠𝑎.
ii.
Determine the measure of ∠𝑏.
iii. Explain how you were able to determine the measure of the unknown angle in part (i) or
part (ii).
Module 2:
Similarity, Proof, and Trigonometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M2
GEOMETRY
d.
A ball is dropped from the top of a 45 ft. building. Once the ball is released, a strong gust of wind
blew the ball off course, and it dropped 4 ft. from the base of the building.
i.
Sketch a diagram of the situation.
ii.
By approximately how many degrees was the ball blown off course? Round your answer to the
nearest whole degree.
Module 2:
Similarity, Proof, and Trigonometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M2
GEOMETRY
̅̅̅̅ in the figure below. Point 𝐴 is
3. A radio tower is anchored by long cables called guy wires, such as 𝐴𝐵
250 m from the base of the tower, and 𝑚∠𝐵𝐴𝐶 = 59°.
a.
How long is the guy wire? Round to the nearest tenth.
b.
How far above the ground is it fastened to the tower?
c.
̅̅̅̅ , if 𝑚∠𝐷𝐴𝐶 = 71˚?
How tall is the tower, 𝐷𝐶
Module 2:
Similarity, Proof, and Trigonometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M2
GEOMETRY
4. The following problem is modeled after a surveying question developed by a Chinese mathematician
during the Tang dynasty in the seventh century C.E.
A building sits on the edge of a river. A man views the building from the opposite side of the river. He
measures the angle of elevation with a handheld tool and finds the angle measure to be 45°. He moves
50 feet away from the river and remeasures the angle of elevation to be 30°.
What is the height of the building? From his original location, how far away is the viewer from the top of
the building? Round to the nearest whole foot.
Module 2:
Similarity, Proof, and Trigonometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M2
GEOMETRY
5. Prove the Pythagorean theorem using similar triangles. Provide a well-labeled diagram to support your
justification.
Module 2:
Similarity, Proof, and Trigonometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M2
GEOMETRY
̅̅̅̅ with the hypotenuse so
6. In right triangle 𝐴𝐵𝐶 with ∠𝐵 a right angle, a line segment 𝐵′𝐶′ connects side 𝐴𝐵
that ∠𝐴𝐵′𝐶′ is a right angle as shown. Use facts about similar triangles to show why cos 𝐶′ = cos 𝐶 .
Module 2:
Similarity, Proof, and Trigonometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M2
GEOMETRY
7. Terry said, “I will define the zine of an angle 𝑥 as follows. Build an isosceles triangle in which the sides of
equal length meet at angle 𝑥. The zine of 𝑥 will be the ratio of the length of the base of that triangle to
the length of one of the equal sides.” Molly said, “Won’t the zine of 𝑥 depend on how you build the
isosceles triangle?”
a.
What can Terry say to convince Molly that she need not worry about this? Explain your answer.
b.
Describe a relationship between zine and sin.
Module 2:
Similarity, Proof, and Trigonometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M3
GEOMETRY
Name
Date
1.
a.
State the volume formula for a cylinder. Explain why the volume formula works.
b.
The volume formula for a pyramid is 𝐵ℎ, where 𝐵 is the area of the base and ℎ is the height of the
1
1
3
solid. Explain where the comes from in the formula.
3
Module 3:
Extending to Three Dimensions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M3
GEOMETRY
c.
Explain how to use the volume formula of a pyramid to show that the volume formula of a circular
1
cone is 𝜋𝑟2 ℎ, where 𝑟 is the radius of the cone and ℎ is the height of the cone.
3
2. A circular cylinder has a radius between 5.50 and 6.00 cm and a volume of 225 cm3 . Write an inequality
that represents the range of possible heights the cylinder can have to meet this criterion to the nearest
hundredth of a centimeter.
Module 3:
Extending to Three Dimensions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M3
GEOMETRY
3. A machine part is manufactured from a block of iron with circular cylindrical slots. The block of iron has a
width of 14 in., a height of 16 in., and a length of 20 in. The number of cylinders drilled out of the block
is determined by the weight of the leftover block, which must be less than 1,000 lb.
a.
If iron has a weight of roughly 491 lb/ft 3, how many cylinders with the same height as the block and
with radius 2 in. must be drilled out of the block in order for the remaining solid to weigh less than
1,000 lb.?
b.
If iron ore costs $115 per ton (1 ton = 2200 lb.) and the price of each part is based solely on its
weight of iron, how many parts can be purchased with $1,500? Explain your answer.
Module 3:
Extending to Three Dimensions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M3
GEOMETRY
4. Rice falling from an open bag piles up into a figure conical in shape
with an approximate radius of 5 cm.
a.
If the angle formed by the slant of the pile with the base is
roughly 30°, write an expression that represents the volume of
rice in the pile.
b.
If there are approximately 20 grains of rice in a cubic centimeter, approximately how many grains of
rice are in the pile? Round to the nearest whole grain.
Module 3:
Extending to Three Dimensions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M3
GEOMETRY
5. In a solid hemisphere, a cone is removed as shown. Calculate the volume of the resulting solid. In
addition to your solution, explain the strategy you used in your solution.
6. Describe the shape of the cross-section of each of the following objects.
Right circular cone:
a. Cut by a plane through the vertex and perpendicular to the base
Square pyramid:
b. Cut by a plane through the vertex and perpendicular to the base
c.
Cut by a vertical plane that is parallel to an edge of the base but not passing through the vertex
Module 3:
Extending to Three Dimensions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M3
GEOMETRY
Sphere with radius 𝑟:
d. Describe the radius of the circular cross-section created by a plane through the center of the sphere.
e.
Describe the radius of the circular cross-section cut by a plane that does not pass through the center
of the sphere.
Triangular Prism:
f. Cut by a plane parallel to a base
g.
Cut by a plane parallel to a face
a.
A 3 × 5 rectangle is revolved about one of its sides of length 5 to create a solid of revolution. Find
the volume of the solid.
7.
Module 3:
Extending to Three Dimensions
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M3
GEOMETRY
b.
A 3-4-5 right triangle is revolved about a leg of length 4 to create a solid of revolution. Describe the
solid.
c.
A 3-4-5 right triangle is revolved about its legs to create two solids. Find the volume of each solid
created.
d.
Show that the volume of the solid created by revolving a 3-4-5 triangle about its hypotenuse is
Module 3:
Extending to Three Dimensions
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5
𝜋.
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M4
GEOMETRY
Name
Date
For problems that require rounding, round answers to the nearest hundredth.
1. Given parallelogram 𝑅𝑆𝑇𝑈 with vertices 𝑅(1, 3), 𝑆(−2, −1), 𝑇(4, 0), and 𝑈(7, 4):
a.
Find the perimeter of the parallelogram; round to
the nearest hundredth.
b.
Find the area of the parallelogram.
2. Given triangle 𝐴𝐵𝐶 with vertices 𝐴(6, 0), 𝐵(−2, 2), and 𝐶(−3, −2):
a.
Find the perimeter of the triangle; round to the
nearest hundredth.
b.
Find the area of the triangle.
Module 4:
Connecting Algebra and Geometry Through Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M4
GEOMETRY
3. A triangular region in the coordinate plane is defined by the system of inequalities
1
𝑦 ≥ 𝑥 − 6, 𝑦 ≤ −2𝑥 + 9, 𝑦 ≤ 8𝑥 + 9.
2
a.
Determine the coordinates of the vertices or the triangle.
b.
Sketch the triangular region defined by these inequalities.
c.
Is the triangle defined by the inequalities a right triangle? Explain your answer.
Module 4:
Connecting Algebra and Geometry Through Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M4
GEOMETRY
d.
Find the perimeter of the triangular region defined by the inequalities; round to the nearest
hundredth.
e.
What is the area of this triangular region?
f.
Of the three altitudes of the triangular region defined by the inequalities, what is the length of the
shortest of the three? Round to the nearest hundredth.
4. Find the point on the directed line segment from (0, 3) to (6, 9) that divides the segment in the ratio of
2: 1.
Module 4:
Connecting Algebra and Geometry Through Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M4
GEOMETRY
5. Consider the points 𝐴(1, 4) and 𝐵(8, −3). Suppose 𝐶 and 𝐷 are points on the line through 𝐴 and 𝐵
satisfying
𝐴𝐶
𝐶𝐵
=
1
3
and
𝐵𝐷
𝐷𝐴
4
= , respectively.
3
a.
Draw a sketch of the four collinear points 𝐴, 𝐵, 𝐶, and 𝐷, showing their relative positions to one
another.
b.
Find the coordinates of point 𝐶.
c.
Find the coordinates of point 𝐷.
Module 4:
Connecting Algebra and Geometry Through Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M4
GEOMETRY
6. Two robots are left in a robotics competition. Robot A is programmed to move about the coordinate
plane at a constant speed so that, at time 𝑡 seconds, its position in the plane is given by
𝑡
(0, 10) + (60, 80).
8
Robot B is also programmed to move about the coordinate plane at a constant speed. Its position in the
plane at time 𝑡 seconds is given by
𝑡
(70, 0) −
(70, −70).
10
a.
What was each robot’s starting position?
b.
Where did each robot stop?
c.
What is the equation of the path of robot A?
d.
What is the equation of the path of robot B?
Module 4:
Connecting Algebra and Geometry Through Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M4
GEOMETRY
e.
What is the speed of robot A? (Assume coordinates in the plane are given in units of meters. Give
the speed in units of meters per second.)
f.
Do the two robots ever pass through the same point in the plane? Explain. If they do, do they pass
through that common point at the same time? Explain.
g.
What is the closest distance robot B will ever be to the origin? Round to the nearest hundredth.
h.
At time 𝑡 = 10, robot A will instantaneously turn 90 degrees to the left and travel at the same
constant speed it was previously traveling. What will be its coordinates in another 10 seconds’
time?
Module 4:
Connecting Algebra and Geometry Through Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M4
GEOMETRY
7.
𝐺𝐷𝐴𝑌 is a rhombus. If point 𝐺 has coordinates (2, 6) and 𝐴 has coordinates (8, 10), what is the
equation of the line that contains the diagonal ̅̅̅̅
𝐷𝑌 of the rhombus?
8.
a.
A triangle has vertices 𝐴(𝑎1 , 𝑎2 ), 𝐵(𝑏1 , 𝑏2 ), and 𝐶(𝑐1 , 𝑐2 ). Let 𝑀 be the midpoint of ̅̅̅̅
𝐴𝐶 and 𝑁 the
̅̅̅̅ . Find a general expression for the slope of ̅̅̅̅̅
midpoint of 𝐵𝐶
𝑀𝑁. What segment of the triangle has
the same slope as ̅̅̅̅̅
𝑀𝑁?
b.
̅̅̅̅ with
A triangle has vertices 𝐴(𝑎1 , 𝑎2 ), 𝐵(𝑏1 , 𝑏2 ), and 𝐶(𝑐1 , 𝑐2 ). Let 𝑃 be a point on 𝐴𝐶
5
5
𝐴𝑃 = 8 𝐴𝐶, and let 𝑄 be a point on ̅̅̅̅
𝐵𝐶 with 𝐵𝑄 = 8 𝐵𝐶. Find a general expression for the slope of
̅̅̅̅
𝑃𝑄 . What segment of the triangle has the same slope as ̅̅̅̅
𝑃𝑄 ?
Module 4:
Connecting Algebra and Geometry Through Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M4
GEOMETRY
c.
A quadrilateral has vertices 𝐴(𝑎1 , 𝑎2 ), 𝐵(𝑏1 , 𝑏2 ), 𝐶(𝑐1 , 𝑐2 ), and 𝐷(𝑑1 , 𝑑2 ). Let 𝑅, 𝑆, 𝑇, and 𝑈 be the
̅̅̅̅ is parallel to ̅̅̅̅
midpoints of the sides ̅̅̅̅
𝐴𝐵 , ̅̅̅̅
𝐵𝐶 , ̅̅̅̅
𝐶𝐷, and ̅̅̅̅
𝐷𝐴, respectively. Demonstrate that 𝑅𝑆
𝑇𝑈.
̅
̅
̅̅
̅̅̅̅
Is 𝑆𝑇 parallel to 𝑈𝑅? Explain.
9. The Pythagorean theorem states that if three squares are drawn on the
sides of a right triangle, then the area of the largest square equals the sum
of the areas of the two remaining squares.
There must be a point 𝑃 along the hypotenuse of the right triangle at which
the large square is divided into two rectangles as shown, each with an area
matching the area of one of the smaller squares.
Module 4:
Connecting Algebra and Geometry Through Coordinates
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End-of-Module Assessment Task
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
GEOMETRY
Consider a right triangle 𝐴𝑂𝐵 situated on the coordinate plane with vertex 𝐴 on the positive 𝑦-axis, 𝑂 at
the origin, and vertex 𝐵 on the positive 𝑥-axis.
̅̅̅̅ is 𝑐.
Suppose 𝐴 has coordinates (0, 𝑎), 𝐵 has coordinates (𝑏, 0), and the length of the hypotenuse 𝐴𝐵
a.
Find the coordinates of a point 𝑃 on ̅̅̅̅
𝐴𝐵 such that ̅̅̅̅
𝑂𝑃 is
̅̅̅̅.
perpendicular to 𝐴𝐵
b.
Show that for this point 𝑃 we have
c.
Show that if we draw from 𝑃 a line perpendicular to ̅̅̅̅
𝐴𝐵 , then that line divides the square with ̅̅̅̅
𝐴𝐵 as
2
one of its sides into two rectangles, one of area 𝑎 and one of area 𝑏 2 .
Module 4:
𝐴𝑃
𝑃𝐵
=
𝑎2
𝑏2
.
Connecting Algebra and Geometry Through Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M5
GEOMETRY
Name
Date
1. Let 𝐶 be the circle in the coordinate plane that passes though the points (0, 0), (0, 6), and (8, 0).
a.
What are the coordinates of the center of the circle?
b.
What is the area of the portion of the interior of the circle that lies in the first quadrant? (Give an
exact answer in terms of 𝜋.)
Module 5:
Circles With and Without Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M5
GEOMETRY
c.
What is the area of the portion of the interior of the circle that lies in the second quadrant? (Give an
approximate answer correct to one decimal place.)
d.
What is the length of the arc of the circle that lies in the first quadrant with endpoints on the axes?
(Give an exact answer in terms of 𝜋.)
e.
What is the length of the arc of the circle that lies in the second quadrant with endpoints on the
axes? (Give an approximate answer correct to one decimal place.)
Module 5:
Circles With and Without Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M5
GEOMETRY
f.
A line of slope −1 is tangent to the circle with point of contact in the first quadrant. What are the
coordinates of that point of contact?
g.
Describe a sequence of transformations that show circle 𝐶 is similar to a circle with radius one
centered at the origin.
Module 5:
Circles With and Without Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M5
GEOMETRY
h.
If the same sequence of transformations is applied to the tangent line described in part (f), will the
image of that line also be a line tangent to the circle of radius one centered about the origin? If so,
what are the coordinates of the point of contact of this image line and this circle?
2. In the figure below, the circle with center 𝑂 circumscribes △ 𝐴𝐵𝐶.
Points 𝐴, 𝐵, and 𝑃 are collinear, and the line through 𝑃 and 𝐶 is tangent to the circle at 𝐶. The center of
the circle lies inside △ 𝐴𝐵𝐶.
a.
Find two angles in the diagram that are congruent, and explain why they are congruent.
Module 5:
Circles With and Without Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M5
GEOMETRY
b.
̅̅̅̅ and 𝑃𝐶 = 7, what is the length of ̅̅̅̅
If 𝐵 is the midpoint of 𝐴𝑃
𝑃𝐵?
c.
If 𝑚∠𝐵𝐴𝐶 = 50°, and the measure of the arc 𝐴𝐶 is 130°, what is 𝑚∠𝑃?
Module 5:
Circles With and Without Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M5
GEOMETRY
3. The circumscribing circle and the inscribed circle of a triangle have the same center.
a.
By drawing three radii of the circumscribing circle, explain why the triangle must be equiangular and,
hence, equilateral.
Module 5:
Circles With and Without Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M5
GEOMETRY
b.
Prove again that the triangle must be equilateral, but this time by drawing three radii of the
inscribed circle.
c.
Describe a sequence of straightedge and compass constructions that allows you to draw a circle
inscribed in a given equilateral triangle.
Module 5:
Circles With and Without Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M5
GEOMETRY
4.
a.
Show that
(𝑥 − 2)(𝑥 − 6) + (𝑦 − 5)(𝑦 + 11) = 0
is the equation of a circle. What is the center of this circle? What is the radius of this circle?
b.
A circle has diameter with endpoints (𝑎, 𝑏) and (𝑐, 𝑑). Show that the equation of this circle can be
written as
(𝑥 − 𝑎)(𝑥 − 𝑐) + (𝑦 − 𝑏)(𝑦 − 𝑑) = 0.
Module 5:
Circles With and Without Coordinates
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
M5
GEOMETRY
5. Prove that opposite angles of a cyclic quadrilateral are supplementary.
Module 5:
Circles With and Without Coordinates
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