Two pages of visual notes for students to keep: the key words with pictures, how to tell an enlargement from a reduction, four solved problems in the style of the Similarity Builder game, and where similar figures show up in real life. To make a PDF, click Print and choose "Save as PDF."
Visual Notes: Similar Figures and Scale Factor
Grades 7–8 · CCSS 7.G.A.1, 8.G.A.4 · Pairs with the Similarity Builder game
Name
The big idea: similar figures are the same shape but not always the same size, like a photo and its enlargement. Every length is multiplied by the same number, the scale factor k, and every angle stays the same. Multiply to grow or shrink; never add.
Key vocabulary
Similar figures
Same shape: equal angles and proportional sides.
Dilation
A transformation that enlarges or shrinks a figure by a scale factor.
Scale factor (k)
k = image side ÷ original side
For 2 × 3 → 4 × 6: k = 4 ÷ 2 = 2.
Corresponding sides
Sides in matching positions (same color). Each pair has the same ratio, here 2.
Corresponding angles
Angles in matching positions. In similar figures they are equal.
Proportional
small
3
4
big
6
8
Every pair of matching sides has the same ratio: 6/3 = 8/4 = 2.
Grow, shrink or stay the same?
k > 1: enlargementThe image is bigger. (k = 2)
k < 1: reductionThe image is smaller. (k = ½)
k = 1: same sizeThe figures are congruent.
Two ways to use k: image side = k × original side, so original side = image side ÷ k. Always match sides in the same position before you divide.
Visual Notes: Similar Figures and Scale Factor, page 2
1. Find the scale factor
Match sides: 3 → 9 and 5 → 15.
k = image ÷ original = 9 ÷ 3 = 3
It's more than 1, so it's an enlargement.
✓ Check: the other pair agrees, 15 ÷ 5 = 3
2. Use the scale factor
k = 2.5. The original side is 6. Find the image side.
Image side = k × original side
= 2.5 × 6 = 15
✓ Check: the other side, 4 × 2.5 = 10, matches the picture
3. Find k first, then the missing side
Matching widths: 6 → 4, so k = 4 ÷ 6 = ⅔ (a reduction).
Missing side = ⅔ × 9 = 6
✓ Check: 4/6 = 6/9 = ⅔
4. Real life: shadows
A 6 ft person casts a 4 ft shadow. At the same time a tree casts an 18 ft shadow. How tall is the tree?
The sun makes similar triangles: k = 18 ÷ 4 = 4.5
Tree = 4.5 × 6 = 27 ft
✓ Check: 6/4 = 27/18 = 1.5
Watch out
Multiply, don't add. If 3 grows to 9 (× 3), then 4 grows to 12, not 4 + 6 = 10.
Divide image by original (new ÷ old) to get k. Upside down gives the wrong k.
Angles never get multiplied. A 40° angle is still 40° after any dilation.
Where you see it in real life
Photos: enlarging keeps the same shape.Maps: a scale shrinks real distances by the same factor.Shadows: the sun makes similar triangles.Screens: TVs of different sizes have the same shape.