Dilations Ws 2 | PDF - Free Printable

Let’s solve each problem step by step.

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Problem 1: k = 2

We are dilating triangle ABC with scale factor 2, centered at the origin (unless otherwise specified — and since no center is given, we assume origin).

First, find coordinates of points:

- A is at (1, 1)
- B is at (3, -1)
- C is at (-2, 0)

Multiply each coordinate by 2:

- A’ = (2, 2)
- B’ = (6, -2)
- C’ = (-4, 0)

Draw new triangle using these points.

✔ Answer for #1: New vertices: A’(2,2), B’(6,-2), C’(-4,0)

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Problem 2: k = 1/4

Triangle ABC:

- A is at (0, 3)
- B is at (3, -3)
- C is at (-3, -3)

Multiply each coordinate by 1/4:

- A’ = (0, 3/4) → (0, 0.75)
- B’ = (3/4, -3/4) → (0.75, -0.75)
- C’ = (-3/4, -3/4) → (-0.75, -0.75)

✔ Answer for #2: New vertices: A’(0, 0.75), B’(0.75, -0.75), C’(-0.75, -0.75)

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Problem 3: k = 1/2

Figure ABCD:

- A is at (3, 2)
- B is at (3, 5)
- C is at (5, 2)
- D is at (5, 4)

Multiply each coordinate by 1/2:

- A’ = (1.5, 1)
- B’ = (1.5, 2.5)
- C’ = (2.5, 1)
- D’ = (2.5, 2)

✔ Answer for #3: New vertices: A’(1.5,1), B’(1.5,2.5), C’(2.5,1), D’(2.5,2)

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Problem 4: k = 1½ = 3/2

Quadrilateral ABCD:

- A is at (2, 0)
- B is at (0, -2)
- C is at (-2, 1)
- D is at (1, 2)

Multiply each coordinate by 3/2:

- A’ = (2 × 3/2, 0 × 3/2) = (3, 0)
- B’ = (0 × 3/2, -2 × 3/2) = (0, -3)
- C’ = (-2 × 3/2, 1 × 3/2) = (-3, 1.5)
- D’ = (1 × 3/2, 2 × 3/2) = (1.5, 3)

✔ Answer for #4: New vertices: A’(3,0), B’(0,-3), C’(-3,1.5), D’(1.5,3)

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Problem 5: Figure A to Figure B

Figure A is a rectangle: width = 6, height = 3
Figure B is a smaller rectangle: width = y, height = 2

Since it's a dilation, ratios must be equal.

Compare heights:
Original height = 3 → New height = 2
Scale factor = 2/3

So width should also shrink by 2/3:
Original width = 6 → New width = 6 × (2/3) = 4 → so y = 4

Also, side x corresponds to original side 2? Wait — look again.

Actually, in Figure B, sides are labeled: top = y, left = x, right = z, bottom = 2? That doesn’t make sense unless it’s rotated.

Wait — better interpretation:

Figure A: rectangle 6 wide, 3 tall
Figure B: rectangle ? wide, 2 tall — and labeled with variables on sides.

Assuming corresponding sides:

Height: 3 → 2 ⇒ scale factor = 2/3 < 1 ⇒ reduction

Width: 6 → y ⇒ y = 6 × (2/3) = 4

Now, what about x and z? In Figure B, if it’s drawn as a small rectangle with height 2 and width y=4, then:

If x is the left side, that’s height → x = 2
z is the right side → also height → z = 2
But wait — maybe labels are different.

Looking at diagram description: “x B z” above, “2” below — probably means:

Top side = y
Left side = x
Right side = z
Bottom side = 2

But in a rectangle, opposite sides equal → so x = z, and y = 2? No — that contradicts.

Wait — perhaps Figure B has dimensions: vertical sides = x and z, horizontal sides = y and 2.

In a rectangle, opposite sides equal → so x = z, and y = 2? But then why label both?

Alternatively, maybe it’s not aligned — but likely, the 2 is one side, and y is the other.

Given Figure A is 6 by 3, Figure B is scaled version.

If the side labeled “2” in B corresponds to the side of length 3 in A, then scale factor = 2/3.

Then the other side (y) corresponds to 6 → y = 6 * 2/3 = 4

And since it’s a rectangle, the other two sides (x and z) should both equal 2 (if they’re the heights). But in the diagram, it says “x B z” on top — maybe x and z are the vertical sides? Then x = z = 2.

But that would mean all sides are known except y.

Perhaps the labeling is:

- Top: y
- Left: x
- Right: z
- Bottom: 2

In rectangle, top = bottom → y = 2? But that can’t be because then scale factor wouldn’t match.

I think there’s confusion in labeling. Let me re-read:

“5. [diagram] x B z over 2, and A is 6x3 rectangle”

Probably, Figure B is a small rectangle with:

- Height = 2 (given on bottom)
- Width = y (on top)
- Left side = x
- Right side = z

But in rectangle, left = right, top = bottom → so x = z, and y = 2? But then why have y?

Unless... perhaps the 2 is not the bottom side but a label for something else.

Another possibility: the "2" is the length of the side corresponding to the 3 in A.

So:

Corresponding sides:

A: 3 (height) → B: 2 ⇒ scale factor = 2/3

A: 6 (width) → B: y ⇒ y = 6 * 2/3 = 4

Then, since it’s a rectangle, the other sides (x and z) should be equal to the height, which is 2. So x = 2, z = 2.

But in the diagram, it might be that x and z are the widths? Unlikely.

Perhaps the variable labels are on the sides, and we need to assign based on correspondence.

To avoid confusion, let’s assume:

- The side of length 3 in A corresponds to the side of length 2 in B → scale factor 2/3
- The side of length 6 in A corresponds to side y in B → y = 4
- The other two sides in B are the same as the 2-side, so x = 2, z = 2

But typically in such diagrams, if it's labeled "x B z" on top, it might mean the top side is divided or something — but I think it's safe to say:

Scale factor = 2/3 (reduction)

y = 4

x = 2 (left side, same as height)

z = 2 (right side)

But let's check problem 6 for pattern.

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Problem 6: Triangles A and B

Triangle A: sides 7, 8, 10
Triangle B: sides 14, m, n

Clearly, 14 / 7 = 2, so scale factor = 2 (enlargement)

Then m = 8 * 2 = 16
n = 10 * 2 = 20

✔ Answer for #6: enlargement, m=16, n=20

Back to #5: similarly, if Figure A is 6x3, Figure B has a side of 2, which likely corresponds to the 3-side, so scale factor 2/3, reduction.

Then the other dimension: 6 * 2/3 = 4 = y

And the sides perpendicular: since it's a rectangle, the height is 2, so x and z should be 2 if they are the vertical sides.

But in the diagram description, it says "x B z" above the rectangle, and "2" below — perhaps "2" is the bottom side, so bottom = 2, top = y, left = x, right = z.

In rectangle, top = bottom → y = 2? But that contradicts scaling.

Unless the 2 is not a side length but a label — but it's written as "2", same as in A it's "3" and "6".

I think there's a misinterpretation.

Another idea: perhaps Figure B is oriented differently, and the side labeled "2" corresponds to the side of length 6 in A.

Let's try that.

If side 6 in A corresponds to side 2 in B, then scale factor = 2/6 = 1/3

Then side 3 in A corresponds to y in B → y = 3 * 1/3 = 1

Then x and z would be the other sides, which should be 1 each? But then why label them separately.

This is ambiguous.

Look at the diagram description: "x B z" on top, "2" on bottom — perhaps it's not a rectangle but a different shape? But it says "rectangle" implicitly.

Perhaps "B" is the figure, and x,y,z are variables on its sides.

Standard way: in dilation, corresponding sides proportional.

Assume that the side of length 3 in A corresponds to the side of length 2 in B. Why? Because 2 is given, and 3 is the smaller side in A, likely corresponding.

So scale factor k = 2/3

Then the side of length 6 in A corresponds to y in B → y = 6 * 2/3 = 4

Now, for x and z: in Figure B, if it's a rectangle with width y=4 and height 2, then the left and right sides are both 2, so x=2, z=2.

Perhaps the labeling "x B z" means that the top side is composed of x and z or something, but that seems unlikely.

I think for consistency, we'll go with:

- Reduction (since 2<3)
- y = 4
- x = 2
- z = 2

But let's see problem 7 for more clues.

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Problem 7: Find scale factor, reduction/enlargement, and variables.

There are two subproblems.

Subproblem 1:

Two triangles sharing vertex C.

Small triangle: sides 4,5,6? Labels: P'C = 4, PP' = 5, PC = 6? Not clear.

Diagram: large triangle with points P', P, C.

Sides: from C to P' is 4, C to P is 6, P' to P is 5? But 4,5,6 don't form a triangle with those lengths? 4+5>6, ok.

Then larger triangle has sides x, y, and from C to some point.

Actually, it says: "P' 5 P C" and "4" on CP', "6" on CP, and "12" on another side.

Read: "Find the scale factor. Tell whether the dilation is a reduction or an enlargement. Then find the values of the variables."

Diagram: two triangles, one inside the other, sharing vertex C.

Small triangle: vertices C, P', P

Sides: CP' = 4, CP = 6, P'P = 5

Large triangle: vertices C, and two other points, with sides x, y, and 12.

Specifically, it shows: from C to a point is 12, and that corresponds to CP=6? Or CP'=4?

Typically, in such diagrams, the correspondence is along the rays from center C.

So, ray CP: small segment CP=6, large segment is say CQ=12, so scale factor = 12/6 = 2

Similarly, ray CP': small CP'=4, large should be 4*2=8, but not labeled.

The large triangle has side x corresponding to P'P=5, so x=5*2=10

Side y corresponding to... what? The base.

In small triangle, side P'P=5, in large triangle, the corresponding side is x, so x=10

Then y is another side, perhaps corresponding to CP' or something.

The diagram shows "y" on the base of the large triangle, and "x" on the side.

Also, there is "12" on the side from C to the far point, which corresponds to CP=6, so scale factor k=12/6=2

Then, the side corresponding to CP'=4 should be 8, but not asked.

The side corresponding to P'P=5 is x, so x=10

Now, what is y? In the large triangle, the base is y, which should correspond to the base of the small triangle, which is P'P=5, so y=10? But x is already 10.

Perhaps y is the other side.

Looking at the diagram description: "x" on one side, "y" on the base, and "12" on the side from C.

Also, there is "6" on CP, "4" on CP', "5" on P'P.

In large triangle, from C to the end of the base is 12, which is along CP, so corresponds to CP=6, so k=2.

Then the side from C to the other end of the base should correspond to CP'=4, so 8.

The base of the large triangle corresponds to P'P=5, so should be 10.

But in the diagram, the base is labeled y, and one side is labeled x.

Probably x is the side corresponding to P'P, so x=10

y is the base, which is the same as x? Or perhaps y is the other side.

The diagram might have y as the length of the base, which is 10, and x as another side.

But it says "find the values of the variables", and variables are x and y.

Perhaps in the large triangle, the sides are: from C to first point: 12 (corresponds to CP=6), from C to second point: let's call it w, corresponds to CP'=4, so w=8, and the base between them is y, corresponds to P'P=5, so y=10.

But x is labeled on one of the sides — perhaps x is w, so x=8.

The description says: "x" and "y" are variables, and in the diagram, "x" is on the side that is not 12, and "y" on the base.

Assume:

- Side from C to P in small triangle: 6
- Corresponding side in large triangle: 12 → k=2

- Side from C to P' in small triangle: 4
- Corresponding side in large triangle: x → x=4*2=8

- Side P'P in small triangle: 5
- Corresponding side in large triangle: y → y=5*2=10

Yes, that makes sense.

So scale factor = 2, enlargement, x=8, y=10

✔ Answer for 7.1: scale factor 2, enlargement, x=8, y=10

Subproblem 2:

Rectangles.

Outer rectangle: width 12, height 6

Inner rectangle: width 5, height ? , and labeled P', C, x, F10

Diagram: outer rectangle 12 by 6

Inner rectangle inside, with width 5 (labeled on top), and height x (labeled on side), and point C, P', etc.

Also, "F10" might be a typo or label.

Probably, the inner rectangle is a dilation of the outer or vice versa.

Typically, if inner is smaller, it might be reduction.

Assume the inner rectangle is the image of the outer under dilation.

But usually, dilation from a center.

Here, likely centered at a corner or center.

Notice that the inner rectangle has width 5, outer has width 12.

If corresponding, scale factor k = 5/12

Then height of inner should be 6 * (5/12) = 30/12 = 2.5

But it's labeled x, so x=2.5

Is it reduction or enlargement? Since 5<12, reduction.

But let's confirm correspondence.

The diagram shows "5" on the top of inner rectangle, "12" on top of outer, so yes, widths correspond.

Heights: outer is 6, inner is x, so x = 6 * (5/12) = 30/12 = 5/2 = 2.5

Also, there is "C" and "P'" — probably not needed for calculation.

"F10" might be irrelevant.

So scale factor = 5/12, reduction, x=2.5

But 2.5 is 5/2, or 2.5.

We can write as fraction.

✔ Answer for 7.2: scale factor 5/12, reduction, x=5/2 or 2.5

Now back to problem 5.

In problem 5, Figure A is 6 by 3, Figure B has a side of 2.

If we assume that the side of length 3 in A corresponds to the side of length 2 in B, then k=2/3, reduction.

Then the other side: 6 * 2/3 = 4 = y

Now for x and z: in Figure B, if it's a rectangle with width y=4 and height 2, then the left and right sides are both 2, so if x and z are those, then x=2, z=2.

Perhaps the labeling "x B z" means that the top side is y, and the left and right are x and z, which are both equal to the height, 2.

So I'll go with that.

For problem 5: reduction, y=4, x=2, z=2

But let's write answers clearly.

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Final Answers:

1. Dilated vertices: A'(2,2), B'(6,-2), C'(-4,0)

2. Dilated vertices: A'(0, 0.75), B'(0.75, -0.75), C'(-0.75, -0.75)

3. Dilated vertices: A'(1.5,1), B'(1.5,2.5), C'(2.5,1), D'(2.5,2)

4. Dilated vertices: A'(3,0), B'(0,-3), C'(-3,1.5), D'(1.5,3)

5. Reduction; y=4, x=2, z=2

6. Enlargement; m=16, n=20

7.1 Scale factor 2, enlargement; x=8, y=10

7.2 Scale factor 5/12, reduction; x=2.5 or 5/2

For problem 5, to be precise, since the diagram might have specific correspondence, but based on standard interpretation, this should be fine.

In problem 5, if "2" is the bottom side, and it corresponds to the bottom side of A which is 6, then k=2/6=1/3, then y (top side) corresponds to top side of A which is 6, so y=2, but then height would be 3*1/3=1, so x and z would be 1.

But that would make Figure B very small, and y=2, but in A, width is 6, so if y=2, it matches, but then why have x and z.

In that case, if bottom=2 corresponds to bottom=6, k=1/3, then height of B = 3 * 1/3 =1, so if x and z are the vertical sides, x=1, z=1, and y=2 (top side).

But in the diagram, it says "x B z" on top, which might imply that the top side is labeled with x and z, but that doesn't make sense for a single side.

Perhaps "x" and "z" are the lengths of the left and right sides, and "y" is the top, "2" is the bottom.

In a rectangle, top=bottom, so y=2, and left=right, so x=z.

Then if bottom=2 corresponds to bottom of A=6, k=1/3, then height of B = 3 * 1/3 =1, so x=1, z=1, y=2.

But then the width is 2, height is 1, while A is 6 by 3, so ratio 2/6=1/3, 1/3=1/3, ok.

But in this case, y=2, which is given as the bottom, but labeled as y on top, so y=2.

Then x and z are the heights, so x=1, z=1.

But earlier I thought y was the width.

This is confusing.

Perhaps in the diagram, the "2" is not a side length but a label for the figure or something, but it's written as "2", same as in A "3" and "6".

Another way: in many such problems, the number given is the length of the corresponding side.

In Figure A, sides are 6 and 3.

In Figure B, one side is given as 2, and others are variables.

Likely, the 2 corresponds to the 3, because 2 and 3 are both the "height" if oriented the same way.

In A, if 6 is width, 3 is height, in B, if 2 is height, then k=2/3, width y=4, and the other sides are the height, so x=2, z=2.

I think that's more reasonable.

To resolve, let's look at the answer format.

For problem 6, it's clear: sides 7,8,10 to 14,m,n, so 14/7=2, so m=16, n=20.

For problem 5, similarly, if we assume the side of length 3 in A corresponds to the side of length 2 in B, then k=2/3, and the side of length 6 corresponds to y, so y=4.

Then for x and z, since it's a rectangle, and if x and z are the other two sides, they should be equal to the height, which is 2, so x=2, z=2.

I'll stick with that.

So final answers:

Final Answer:
1. A'(2,2), B'(6,-2), C'(-4,0)
2. A'(0, 0.75), B'(0.75, -0.75), C'(-0.75, -0.75)
3. A'(1.5,1), B'(1.5,2.5), C'(2.5,1), D'(2.5,2)
4. A'(3,0), B'(0,-3), C'(-3,1.5), D'(1.5,3)
5. Reduction; y=4, x=2, z=2
6. Enlargement; m=16, n=20
7.1 Scale factor 2, enlargement; x=8, y=10
7.2 Scale factor \frac{5}{12}, reduction; x=\frac{5}{2}