A little map of a big park. Same shape. Different size.
Similar figures have the same shape, though their sizes may differ. Matching angles are equal, and every side length is multiplied by the same scale factor. For example, a 3-by-5 rectangle and a 6-by-10 rectangle are similar: both side lengths are doubled.
The scale factor tells what to multiply each original side length by to get the matching copy side. A factor of 3 triples each length; a factor of 1/2 halves it. Find the factor by dividing a copy side by its matching original side, using the same units.
A scale drawing uses one consistent scale for lengths. A scale of 1 inch to 20 feet means each inch on the drawing stands for 20 feet in real life. For a length of x feet represented by 3 inches, write 1/20 = 3/x. Cross-multiply: 1 × x = 20 × 3, so x = 60 feet. Keep drawing lengths on top and real lengths underneath in both ratios.
Try it. Then open the steps.
Example 1 of 5
A small triangle has sides 3, 4, and 5. A similar triangle is scaled by 3. What is the longest side of the big triangle?
| Small | Scale × 3 | Large |
|---|---|---|
| 3 | × 3 | 9 |
| 4 | × 3 | 12 |
| 5 | × 3 | ? |
Answer: 15
Example 2 of 5
Two similar rectangles, a notebook sketch
A small rectangle is 6 by 4. A similar large rectangle has length 15 (matching the 6). What is the missing width?
Answer: 10
Example 3 of 5
A map of the park, 1 inch to 8 miles of trail network
A park map uses the scale 1 inch : 8 miles. Two trailheads (the spots where trails start) are 3 inches apart on the map. How many miles apart are they really?
Answer: 24 miles
Example 4 of 5
A scale drawing of the school field
A drawing of a soccer field is 4 cm by 6 cm. The real field is 90 m along the 6 cm side. How wide is the real field?
Answer: 60 m
Example 5 of 5
Two similar triangular sails at the lake
Triangle ABC is similar to triangle DEF. AB = 6, DE = 9, AC = 10. Find DF (the side that matches AC).
Answer: 15
Easy to miss: Matching the wrong sides. The short side of one figure has to pair with the short side of the other.
Do this: Pair the sides that do the same job. Short with short. Long with long.
Easy to miss: Adding the scale factor, or flipping it. A factor of 3 is not “add 3.”
Do this: Multiply an original length by the scale factor to get the copy. To go back from the copy to the original, divide by that same factor.
24 problems · Camp, Trail, Summit
