Question

(NOTE: The questions listed have pictures that belong to a certain question. Question 1's picture is the first picture, 3's is the third pic, and so on. Question 2 has no picture, however.)

1.) Describe the transformations that will map triangle A to triangle B and illustrate the similarity between the two triangles.
A)rotate 90° clockwise and then translate 6 units down
B)translate 4 units down and rotate 180° about the origin
C)reflect the triangle across the y-axis and translate 4 units down
D)reflect triangle A across the x-axis and then translate 1 unit right

2.)A sphere is dilated by a scale factor of 1.5 to create a new sphere. How does the volume of the new sphere compare with the volume of the original sphere?
A)The volume of the new sphere is 1.5 times the volume of the original sphere.
B)The volume of the new sphere is 3.0 times the volume of the original sphere.
C)The volume of the new sphere is 1.5^2 times the volume of the original sphere.
D)The volume of the new sphere is 1.5^3 times the volume of the original sphere.

3.)If Kite ABCD is reflected across the x-axis, what are the resulting coordinates of point A?
A)(1, -3)
B)(-3, 1)
C)(3, -1)
D)(-3, -1)

4.)The triangle shown is rotated 90° counterclockwise about the origin. What are the new coordinates of point A?
A)(1, 2)
B)(-2, 1)
C)(2, 1)
D)(-2, -1)

Answer 1

1. The correct option is C.

2. The correct option is D.

3. The correct option is B.

4. The correct option is C.

Explanation:

1.

From figure (1) it is clear that the vertices of triangle A are (1,4), (2,6), (-5,6) and vertices of triangle B are (-1,0), (-2,2), (5,2).

If figure reflects across the y-axis, then

`(x,y)\rightarrow (-x,y)`

The vertices after reflection are (-1,4), (-2,6), (5,6).

If figure translated 4 units down, then

`(x,y)\rightarrow (x,y-4)`

The vertices after reflection and translation are (-1,0), (-2,2), (5,2).

Reflect the triangle A across the y-axis and translate 4 units down to get triangle B.

Therefore the correct option is C.

2.

Volume of sphere is

`V=(4)/(3)\pi r^3`

where r is the radius of the sphere.

A sphere is dilated by a scale factor of 1.5 to create a new sphere. So,

Radius of new sphere = 1.5

Volume of new sphere is

`V_1=(4)/(3)\pi (1.5r)^3`

`V_1=(4)/(3)\pi (1.5)^3(r)^3`

`V_1=(1.5)^3[(4)/(3)\pi (r)^3]`

`V_1=(1.5)^3V`

The volume of the new sphere is 1.5^3 times the volume of the original sphere.

Therefore the correct option is D.

3.

If figure reflects across the x-axis, then

`(x,y)\rightarrow (x,-y)`

From the given figure it is clear that the coordinates of point A are(-3,-1). So the coordinates of point A after reflection are

`(-3,-1)\rightarrow (-3,1)`

Therefore the correct option is B.

4.

If figure rotated 90° counterclockwise about the origin, then

`(x,y)\rightarrow (-y,x)`

From the given figure it is clear that the coordinates of point A are(1,-2). So the coordinates of point A after rotation are

`(1,-2)\rightarrow (2,1)`

Therefore the correct option is C.

Answer 2

1. C . . . . the change in orientation means it's not a rotation. Reflection across the x-axis would put the long side at the bottom.

2. D . . . . volumes are related by the cube of the scale factor of linear dimensions

3. B . . . . that reflection changes the sign of the y-coordinate

4. C . . . . for 90° CCW, (x, y) ⇒ (-y, x), so (1, -2) ⇒ (2, 1)