8th Grade · 8.G.A.4

Similar triangles in 8th grade

Similar triangles have the same shape, and one can be larger or smaller than the other, like a photo and its enlargement. In 8th grade, your child learns what similarity means for any flat shape. Two figures are similar when slides, flips, turns and a resize can carry one exactly onto the other.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

What it means, in plain words

Similar figures match in shape but can differ in size. In 8th grade the definition uses moves. Your child can slide, flip and turn a figure, then dilate it. A dilation makes a figure larger or smaller by multiplying every length by the same number, called the scale factor. If the result lands exactly on the second figure, the two figures are similar.

For similar triangles, this means two things. Matching angles are equal, because a dilation never changes an angle. Matching sides all grow or shrink by the same scale factor. A triangle with sides 3, 4 and 5 is similar to a triangle with sides 6, 8 and 10, because every side doubled. A triangle with sides 6, 8 and 11 is not similar to the first one, because the longest side did not double.

Work with similar triangles in 8th grade has two parts. The first is finding the scale factor: divide a side of the new figure by the matching side of the original. The second is describing the moves that turn one figure into the other. For example: dilate by 2 from the origin, then translate 3 units left and 2 units up.

Your child should be able to:

Explain the difference between congruent figures (same shape and same size) and similar figures (same shape, any size)
Find the scale factor from one figure to another by dividing matching sides
Describe a dilation and a sequence of moves that carries one figure onto another
Use the scale factor to find a missing side length

Example

Triangle $ABC$ has sides of 3 cm, 4 cm and 5 cm. Triangle $DEF$ is similar to it, with a shortest side of 6 cm and a longest side of 10 cm. What is the scale factor, and how long is the middle side of $DEF$ ?

Divide a side of the new triangle by the matching side of the first: $6 \div 3 = 2$ .
Check with the longest sides: $10 \div 5 = 2$ . The scale factor is 2.
The middle side is $4 \times 2 = 8$ cm.
The scale factor is 2, and the middle side of triangle $DEF$ is 8 cm long.

Example

Triangle $ABC$ has corners $A(1, 1)$ , $B(4, 1)$ and $C(2, 3)$ . Dilate it by a scale factor of 2 from the origin, then translate it 3 units left and 2 units up. Where are the new corners?

Dilate first by multiplying every coordinate by 2: $(2, 2)$ , $(8, 2)$ and $(4, 6)$ .
Then translate by subtracting 3 from each x-coordinate and adding 2 to each y-coordinate.
The corners become $(-1, 4)$ , $(5, 4)$ and $(1, 8)$ .
The new triangle has corners at $(-1, 4)$ , $(5, 4)$ and $(1, 8)$ . It is similar to triangle $ABC$ and twice as large.

Where children usually get stuck

Your child says a side of 3 that became 6 "grew by 3" and adds 3 to every other side.
Adding is the first way most children think about growing. Similar figures grow by multiplying, so every side gets the same factor, not the same extra length.
What helps. Draw a 2 by 4 rectangle. Make one copy by adding 3 to each side (5 by 7) and one by doubling each side (4 by 8). Only the doubled copy looks like a larger version of the original.
Your child writes a scale factor of $\frac{1}{2}$ when the new figure is twice as large.
The scale factor goes from the first figure to the second, and the division is easy to do the wrong way around.
What helps. Ask first whether the new figure is larger or smaller. Larger means the scale factor must be more than 1, which catches the reversed division.
Your child divides the shortest side of one triangle by the longest side of the other, and the results for different pairs do not agree.
When one triangle is turned or flipped, matching sides are not in the same place on the page.
What helps. Have your child list the sides of each triangle from shortest to longest first. Shortest goes with shortest, middle with middle and longest with longest.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Figure B is a dilation of Figure A. Who correctly found the scale factor from Figure A to Figure B?
$\displaystyle \text{Student 1: } \frac{6}{3} = 2 \text{, so the scale factor is } 2$
$\displaystyle \text{Student 2: } \frac{3}{6} = \frac{1}{2} \text{, so the scale factor is } \frac{1}{2}$
A
Student 1
B
Student 2
C
Both
D
Neither
Show the answer and how to solve it
Student 1 is correct because the scale factor from Figure A to Figure B is $\frac{6}{3} = \frac{8}{4} = 2$ .
1. 1Look at the corresponding sides. Figure A has width 3 and height 4. Figure B has width 6 and height 8.
2. 2Divide each side length of Figure B by the matching side length of Figure A. Both the width ratio and the height ratio equal 2, confirming the scale factor.
3. 3Student 1 computed $\frac{6}{3} = 2$ , correctly placing Figure B's width in the numerator. That is the correct scale factor from Figure A to Figure B.
4. 4Student 2 computed $\frac{3}{6} = \frac{1}{2}$ , dividing Figure A's width by Figure B's width. That reversed ratio gives the scale factor from B to A, not from A to B.
Common mistake. A common mistake is dividing in the wrong order, such as $\frac{3}{6} = \frac{1}{2}$ . That gives the scale factor from B to A. Always place the Figure B measurement in the numerator when finding the scale factor from A to B.
The idea underneath. A scale factor is multiplicative, so you compare corresponding side lengths with a ratio in the correct order, not with a difference.
2On level
Dilate Triangle ABC by a scale factor of 2 about the origin, then translate the result 2 units right and 1 unit down. Draw the final image in the second grid.
Tap or click each corner, then tap the first corner again to close the shape.
Show the answer and how to solve it
The final image is the triangle with vertices (4, 1), (10, 1), and (6, 5). If you plotted those same three points in a different order, it is still the same triangle.
1. 1Read the original vertices from Triangle ABC: A is at (1, 1), B is at (4, 1), and C is at (2, 3).
2. 2A dilation by a scale factor of 2 about the origin doubles each coordinate.
3. 3Now translate the dilated image 2 units right and 1 unit down. Add 2 to each x-coordinate and subtract 1 from each y-coordinate.
4. 4The final image is the triangle with vertices A''(4, 1), B''(10, 1), and C''(6, 5). Plot these three points on the second grid and connect them.
Common mistake. A common mistake is translating first and then dilating. That would give A''(6, 0), B''(12, 0), and C''(8, 4), which is a different image because the order of the transformations matters.
The idea underneath. Each transformation acts on the result of the previous one, so after you dilate the figure, every later move must be applied to the dilated coordinates, not to the original triangle.
3On level
The table shows corresponding side lengths in Figure A and Figure B. What is the scale factor from Figure A to Figure B?
Figure A side length Figure B side length
4 8
7 14
scale factor
Answer format:
Show the answer and how to solve it
Answer. 2
The scale factor is 2 because each side in Figure B is twice the length of the matching side in Figure A.
1. 1Look at the matching side lengths in the table. Because the scale factor goes from Figure A to Figure B, divide each Figure B side length by the matching Figure A side length.
2. 2Using the first row:
3. 3Check the second row to confirm the ratio is the same:
4. 4Both rows give the same ratio, so the scale factor from Figure A to Figure B is 2. Equivalent unsimplified forms such as $\frac{8}{4}$ and $\frac{14}{7}$ are also correct.
Common mistake. A common mistake is reversing the ratio and writing $\frac{1}{2}$ . Since the dilation goes from Figure A to Figure B, use Figure B ÷ Figure A: $\frac{8}{4} = 2$ , not $\frac{4}{8} = \frac{1}{2}$ .
The idea underneath. In a dilation, every pair of corresponding sides changes by the same multiplicative ratio, so the scale factor must be constant across all rows.
4Stretch
The blue quadrilateral WXYZ is Figure A, and the orange quadrilateral W'X'Y'Z' is Figure B. If you dilate Figure A by a scale factor of 2 about the origin, rotate it $90^{\circ}$ counterclockwise about the origin, and then translate it 1 unit right and 3 units down, does it map onto Figure B?
A
Yes
B
No
Show the answer and how to solve it
Yes. After dilating by 2, rotating $90^{\circ}$ counterclockwise, and translating 1 unit right and 3 units down, every vertex of Figure A lands exactly on the matching vertex of Figure B.
1. 1Look at the blue quadrilateral WXYZ and the orange quadrilateral W'X'Y'Z' on the grid.
2. 2First dilate Figure A by a scale factor of 2 about the origin. Multiply each coordinate by 2.
3. 3Next rotate that image $90^{\circ}$ counterclockwise about the origin using the rule $(x, y) \rightarrow (-y, x)$ .
4. 4Then translate 1 unit right and 3 units down. Add 1 to each x-coordinate and subtract 3 from each y-coordinate.
5. 5These final coordinates match the orange vertices exactly: W'(−1, −1), X'(−3, 5), Y'(−9, 3), and Z'(−7, −3). The stated sequence maps Figure A onto Figure B.
Common mistake. A common mistake is stopping after the rotation. At that point the image has the right size and orientation, but its vertices sit at (−2, 2), (−4, 8), (−10, 6), and (−8, 0), which do not match Figure B. The translation of (1, −3) must be applied to finish the sequence.
The idea underneath. To verify a similarity sequence, you must follow the stated transformations in the stated order and compare the final image exactly to the target figure.