Geometry

Triangle Similarity

Lesson

Two triangles are similar if they have the same shape but possibly different sizes. Corresponding angles are equal and corresponding sides are in proportion.

Three similarity shortcuts

AA: two pairs of corresponding angles equal. (The third pair is forced.)
SAS similarity: two pairs of sides in proportion and included angles equal.
SSS similarity: all three pairs of sides in proportion.

Worked example — solving a proportion

Triangles are similar with corresponding sides 4 and 6 on the small triangle, x and 9 on the large.

\frac{4}{6} = \frac{x}{9}

6x = 36 \implies x = 6

Real-world cue — shadows

At the same time of day, an object and its shadow are similar to another object and its shadow. Set up the proportion to find unknown heights.

Similarity scales areas and perimeters

If two similar figures have side-length ratio $k$ , then:

Perimeter ratio is also $k$ .
Area ratio is $k^2$ .

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{Similar triangles, scale 4 to 12. Ratio?}

Problem 2

\text{Ratio 2:5, smaller side 8. Larger?}

Problem 3

\text{AA similarity needs how many equal angle pairs?}

Problem 4

3/x = 6/12. \text{ Find } x.

Practice

Standard problems matching the lesson.

Problem 5

4/6 = x/9. \text{ Find } x.

Problem 6

5/7 = 15/x. \text{ Find } x.

Problem 7

\text{Similar ratio 1:3, larger 27. Smaller?}

Problem 8

\text{Similar ratio 4:7, smaller 12. Larger?}

Problem 9

\text{Sides 6:8:10 vs 9:12:15. Scale factor (large/small)?}

Problem 10

\text{SSS similarity needs how many side-ratio pairs?}

Problem 11

\text{Similar triangle, third angle if other two are } 50^\circ, 70^\circ

Problem 12

\text{Triangles 6,8,10 and 3,4,5. Ratio?}

Problem 13

8/x = 12/15. \text{ Find } x.

Problem 14

\text{5-ft pole casts 3-ft shadow. Tree's 24-ft shadow. Height?}

Problem 15

\text{6-ft person, 4-ft shadow. Tree shadow 20 ft. Tree height?}

Problem 16

\text{Triangles 9,12,15 and 12,x,20. Find x.}

Problem 17

\text{Similar polygons ratio 2:3, smaller perimeter 16. Larger?}

Problem 18

\text{Similar triangles area ratio for scale 1:3?}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

10/x = 15/(x+5). \text{ Find } x.

Problem 20

\text{8-ft pole, 6-ft shadow. Building 30-ft shadow. Height?}

Problem 21

\text{Similar polygons perim ratio 3:4. Area ratio?}

Problem 22

\text{Polygons ratio 5:8. Smaller area 50. Larger?}

Problem 23

\text{Similar triangles 4,5,6 and 8,x,12. Find x.}

Problem 24

\text{Similar triangles ratio 2:7, smaller 14. Larger?}

Problem 25

\text{4-ft girl, 6-ft shadow. Tower shadow 90 ft. Tower height?}

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