math class 678 4-in-1 skill sheets
8.G.A.4 8th Grade Geometry

Dilations & Similarity

Understand that a figure is similar to another if one can be obtained from the other by a sequence of rigid motions and dilations.

How to explain it

The anchor students hold onto: To dilate about the origin by scale factor k: apply (x, y) -> (kx, ky) to every vertex. If k > 1 the image enlarges; if 0 < k < 1 the image reduces. When a sequence of transformations includes a dilation (k ≠ 1), the image is similar to the preimage — same shape, different size.

Students extend dilation thinking into congruence-through-transformations (#93), then compose dilations with rigid motions to establish similarity (#94 Similarity).

Worked examples

Example 1 Point dilation k=2
Dilate (3, 4) by k=2; center O.
Step 1Apply (x, y) -> (2x, 2y).
Step 2(3, 4) -> (6, 8).
AnswerImage: (6, 8).
Example 2 Triangle reduction k=1/2
Dilate XYZ; k=1/2 about origin.
Step 1Apply (x, y) -> (x/2, y/2) to each vertex.
Step 2X(4, 2) -> X'(2, 1); Y(6, 8) -> Y'(3, 4); Z(2, 6) -> Z'(1, 3).
AnswerX'(2, 1); Y'(3, 4); Z'(1, 3).
Example 3 Dilation about (1, 1)
Dilate PQR; k=2 about (1, 1).
Step 1Apply (x,y) -> (1+2(x-1), 1+2(y-1)) for each vertex.
Step 2P(2, 2) -> P'(3, 3); Q(4, 2) -> Q'(7, 3); R(3, 4) -> R'(5, 7).
AnswerP'(3, 3); Q'(7, 3); R'(5, 7).

Common mistakes

What students write Multiplying only the x-coordinate: (3, 4) dilated k=2 gives (6, 4).
The fix The scale factor multiplies BOTH coordinates. The correct image is (6, 8).
Try this A student was asked to dilate point P(4, 3) by scale factor 2 about the origin. The student wrote: "P' = (4 + 2, 3 + 2) = (6, 5)." Describe the student's error. Then find the correct image P'.
What students write Confusing scale factor k with area factor: k=2 doubles lengths but quadruples area.
The fix Scale factor applies to distances, not areas. Each coordinate is multiplied by k, not k squared.
Try this A student dilated triangle ABC with A(2, 4), B(6, 2), and C(4, 6) by scale factor 1/2 about the origin and recorded: A'(1, 4), B'(3, 2), C'(2, 6). Find the student's error and state the correct image.
What students write Thinking the image is CONGRUENT when a dilation is present.
The fix A dilation changes size. When k ≠ 1 appears in the sequence, the image is SIMILAR, not congruent.
Try this A student found the image of △ABC: A(1, 2), B(3, 2), C(2, 4) after dilating by 2 about the origin then reflecting over the x-axis. Dilate ×2: A(2, 4), B(6, 4), C(4, 8). ✓ Reflect over x-axis: A″(1, −2), B″(3, −2), C″(2, −4). ✗ Describe the error. State the correct A″B″C″.
What students write Applying the second op to the original preimage instead of the intermediate image.
The fix Each step acts on the result of the previous. Chain the operations in order.
Try this A student dilated △JKL — J(2, 1), K(4, 1), L(3, 3) — by 2 about the origin, then translated by (1, 1). Step 1: J(4, 2), K(8, 2), L(6, 6). ✓ Step 2: J″(5, 3), K″(9, 3), L″(7, 7). ✓ The student wrote: “Since the image is twice as big, it is congruent to the original.” Identify the conceptual error. State whether J″K″L″ is similar or congruent to JKL.

Teacher tip

Head off the two predictable errors before they happen. First: The scale factor multiplies BOTH coordinates. The correct image is (6, 8). Second: Scale factor applies to distances, not areas. Each coordinate is multiplied by k, not k squared.