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Learning Math Home
Geometry Session 7: Solutions
 
Session 7 Part A Part B Part C Homework
 
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A B C 
Homework

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Solutions for Session 7 Homework

See solutions for Problems: H1 | H2 | H3 | H4 | H5 | H6


Problem H1

a. 

Any three points on the vertical axis will do -- for instance, (0,-3), (0,1), and (0,13). All the points with x-coordinate 0 are on the y-axis.

b. 

Any three points with the y-coordinate of 1 will do -- for instance, (-7,1), (0,1), and (12,1). All the points on this horizontal line have y-coordinate 1.

c. 

Any three points with x-coordinate 3 will do -- for instance, (3,-4), (3,0), and (3,11). A point is on the line v if its x-coordinate is 3. If its x-coordinate is anything other than 3, the point is not on the line.

d. 

The coordinates are (-4,2).

e. 

For instance, (-3,-3), (-2,-2), (0,0), (4,4), (15,15). All of these points are on the line y = x.

<< back to Problem H1


 

Problem H2

a. 

There are infinitely many points between the two lines -- for instance, (-32,-1), (-17,1) (0,0), (33,3), (155,3.5), (1000,3.9).

b. 

There are infinitely many points which are not between the two lines -- for instance, (-32,7), (-17,-15), (0,5), (33,7), (155,4.5), (1000,7.9).

c. 

A point is between the two lines if its y-coordinate is greater than -2 and less than 4.

<< back to Problem H2


 

Problem H3

A

B

C

D

E

F

G

(x,y)

(x + 3,
y - 2)

(-x,y)

(2x,2y)

(x - 1,
y + 2)

(y,-x)

(-y,x)

(2,1)

(5,-1)

(-2,1)

(4,2)

(1,3)

(1,-2)

(-1,2)

(-4,0)

(-1,-2)

(4,0)

(-8,0)

(-5,2)

(0,4)

(0,-4)

(-5,4)

(-2,2)

(5,4)

(-10,8)

(-6,6)

(4,5)

(-4,-5)


a. 

Triangle B is the translation of triangle A 3 units to the right and 2 units down:

b. 

Triangle C is obtained by reflecting triangle A about the vertical axis:

c. 

Triangle D is obtained by stretching triangle A in both x- and y-direction by a factor of 2:

d. 

Triangle E is obtained by shifting triangle A 1 unit to the left and 2 units up:

e. 

Triangle F is obtained by reflecting triangle A about the vertical axis and then about the line y = x. Alternatively, it can also be obtained as a -90° rotation of triangle A about the origin (0,0).

f. 

Triangle G is obtained by reflecting triangle A about the horizontal axis and then about the line y = x. Alternatively, it can also be obtained as a 90° rotation of triangle A about the origin (0,0).

<< back to Problem H3


 

Problem H4

For example, (-2,3) becomes (-2,-3). In general, reflecting (x,y) about the horizontal axis yields (x,-y). In other words, the x-coordinate is unchanged while the y-coordinate is the negative of the original y-coordinate.

<< back to Problem H4


 

Problem H5

For example, (-2,3) becomes (2,3). In general, reflecting (x,y) about the vertical axis yields (-x,y). In other words, the y-coordinate is unchanged while the x-coordinate is the negative of the original x-coordinate.

<< back to Problem H5


 

Problem H6

For example (-2,3) becomes (3,-2). In general, if a point (x,y) is reflected about the line y = x, its new coordinates are (y,x). In other words, what used to be the x-coordinate becomes the y-coordinate, and what used to be the y-coordinate becomes the x-coordinate.

<< back to Problem H6


 

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