Similar triangles examples are solved by matching corresponding vertices, establishing similarity and using one consistent side ratio. If a triangle with sides 3, 4 and 5 is enlarged by a factor of 2, its matching sides become 6, 8 and 10. The angles stay the same. The worked problems below move from that basic pattern to rotated triangles, algebra, shadows and area, so you can see which quantities to compare and why a plausible-looking proportion can still be wrong.
How triangle similarity works
Two triangles are similar when their matching angles are equal and corresponding side lengths share one scale factor. Their position on the page does not matter. A triangle can be rotated or reflected without changing its shape.
The Cosumnes River College similarity notes give three standard ways to establish similarity: AA, SSS and SAS. AA uses two matching angles; SSS uses three proportional side pairs; SAS uses two proportional side pairs and their equal included angle.
Those conditions are reasons to begin calculating, rather than conclusions to infer from appearance. A drawing that looks like a smaller copy is not enough unless the question gives similarity or enough information to prove it.
Write a correspondence before writing a ratio. If triangle ABC is similar to triangle DEF in that order, A matches D, B matches E and C matches F. Therefore AB matches DE, BC matches EF and AC matches DF. The order carries mathematical information.
For more geometry explanations after these examples, browse the geometry learning articles. The aim is to explain a relationship, not merely collect a final number.
Example 1: Find a missing side using a scale factor
Suppose ABC is similar to DEF, with AB = 6 cm, BC = 8 cm and AC = 10 cm. In the second triangle, DE = 9 cm. Find EF and DF.
First identify a complete corresponding pair: AB and DE. The scale factor from the first triangle to the second is 9 divided by 6, or 1.5.
Multiply every matching length by that same factor. EF corresponds to BC, so EF = 8 × 1.5 = 12 cm. DF corresponds to AC, so DF = 10 × 1.5 = 15 cm.
Check all three ratios: 9/6, 12/8 and 15/10 each equal 1.5. This confirms that the proposed lengths use one consistent enlargement.
A common error is to add 3 to every side because 6 becomes 9. That would produce 11 and 13, but similarity scales lengths by multiplication. It does not add the same fixed amount to every side.
Example 2: Use angles when the triangles are rotated
Triangle PQR has angles P = 40°, Q = 60° and R = 80°. Triangle XYZ has angles X = 80°, Y = 40° and Z = 60°. Suppose PQ = 5 cm and YZ = 15 cm. If QR = 7 cm, find ZX.
Match angles before sides. P corresponds to Y, Q to Z and R to X. A correct ordered statement is PQR similar to YZX.
PQ therefore matches YZ, giving scale factor 15/5 = 3 from the first triangle to the second. QR matches ZX, so ZX = 7 × 3 = 21 cm.
Do not pair P with X simply because each is written first in the original names. The equal angles establish the correspondence. Once you rewrite the triangle names in matching order, the side pairings become much easier to audit.
As a check, ask what would happen if QR were the shortest side while ZX were the longest. That would contradict the correspondence of equal shapes. Relative side size is a useful check, although the angle information remains the actual justification.
Example 3: Solve an algebraic side expression
Two similar triangles have corresponding side pairs 4 cm and 10 cm, and 6 cm and (2x + 1) cm. Find x.
Keep the direction consistent: larger triangle divided by smaller triangle. The known ratio is 10/4 = 2.5. Therefore the expression corresponding to 6 must equal 6 × 2.5 = 15.
Set 2x + 1 = 15. Subtract 1 from both sides to obtain 2x = 14, then divide by 2 to get x = 7.
The requested side length is 15 cm; x itself is 7. Check what the question asks you to report. A variable inside a length expression and the length of the side are different answers.
Substitute back: 2(7) + 1 = 15, and 15/6 = 10/4. If substitution breaks the common ratio, revisit either your algebra or your matching sides.
Example 4: Establish similarity using SSS
Triangle A has sides 5, 7 and 9 units. Triangle B has sides 15, 21 and 27 units. Are they similar?
Compare shortest with shortest, middle with middle and longest with longest. The ratios are 15/5 = 3, 21/7 = 3 and 27/9 = 3. All three agree, so SSS establishes similarity.
Now change only the last side of triangle B to 28. The first two ratios still equal 3, but 28/9 does not. These altered triangles are not similar by the given side lengths.
This is why checking two arbitrary side ratios is insufficient when no included angle is known. Two matches can tempt you to stop before you have enough evidence.
Both sets of side lengths also need to describe actual triangles. For example, 2, 3 and 8 cannot form a triangle because the two shorter lengths together do not exceed the longest. Similarity is a relationship between triangles, not simply lists of numbers.
Example 5: Estimate a height from simultaneous shadows
For an idealized measurement, a vertical 1.5 m pole casts a 2 m shadow on level ground. At the same time, a nearby vertical tree casts a 12 m shadow on the same level surface. Estimate the tree's height.
The pole and tree form right triangles with the ground. Under the stated assumptions, sunlight gives the same acute angle, so the triangles are similar by AA.
Pair height with height and shadow with shadow: tree height divided by 1.5 equals 12 divided by 2. The shadow ratio is 6, so the tree height is 1.5 × 6 = 9 m.
Alternatively, set height/shadow equal in each triangle: h/12 = 1.5/2. Multiplying by 12 gives the same result.
This is a classroom model. Uneven ground, a leaning object or measurements made at different times can undermine its assumptions. The answer should be described as an estimate when working outdoors. OpenStax's proportion applications also treats indirect height measurement using similar figures.
Example 6: Separate side scale from area scale
A triangle has a base of 4 cm and perpendicular height of 3 cm, giving area one-half × 4 × 3 = 6 square centimetres. A similar triangle has corresponding base 12 cm. Find its area.
The length scale factor is 12/4 = 3. Its corresponding perpendicular height is therefore 3 × 3 = 9 cm.
Calculate the new area directly: one-half × 12 × 9 = 54 square centimetres. The area ratio is 54/6 = 9, which is the square of the length factor 3.
The calculation explains the square: both base and height are multiplied by 3. Multiplying the old area by only 3 would scale one dimension while forgetting the other.
Keep units visible. Length is measured in centimetres, while area is measured in square centimetres. If an answer to an area question ends in cm rather than cm², check both your notation and your method.
How to do your own
Start by listing the information you actually have. Mark equal angles, parallel lines and known side lengths. If the problem states similarity, use its ordered triangle names. Otherwise, decide whether AA, SSS or SAS can establish it.
Next write a small correspondence table. This takes seconds and prevents the common mistake of changing pairings halfway through a proportion.
| First triangle | Matching item in second triangle |
|---|---|
| Vertex A | Vertex D |
| Side AB | Side DE |
| Side BC | Side EF |
| Side AC | Side DF |
Choose one direction, such as second divided by first, and keep it for every ratio. Solve the resulting equation, then substitute the result into the original lengths. Finally, decide whether you found a variable, a side, a perimeter or an area.
If you get stuck before the ratio, your difficulty may be angle reasoning rather than arithmetic. The exterior-angle explanation can help review one related tool, although it does not replace a similarity condition.
Try creating a fresh problem by taking a valid triangle and multiplying every side by 1.4. Hide one of the new lengths and recover it using the other pair. Then rotate your sketch and repeat. Your calculation should survive the changed orientation.
How Photo Solver can help
Photo Solver is an iPhone geometry app whose listed features include shape recognition, step-by-step explanations and visual solutions. It can support review when you need help understanding a geometry question after making your own attempt.
For similarity practice, compare an explanation with your correspondence table and identify the exact reason a side pair belongs together. Check important results against your textbook or teacher's solution. Follow your school's rules for AI assistance, especially on assessed work. A photographed answer is most useful when you can close it and explain the same reasoning independently on a fresh question.
Frequently asked questions
How do you find missing sides in similar triangles?
Identify one complete pair of corresponding sides and divide the second length by the first to find a scale factor. Multiply another side of the first triangle by that same factor to find its matching side. Check that you kept the same direction.
Does the direction a triangle faces matter?
No. A rotation or reflection changes how a triangle appears on the page but does not change its shape. Match equal angles or use the ordered similarity statement to establish correspondence before you compare side lengths.
Are two matching side ratios enough to prove similarity?
Not on their own. SSS requires all three corresponding side ratios. SAS requires two proportional side pairs and equality of the included angles. AA provides another route when two corresponding angles are equal.
Is area multiplied by the same scale factor as length?
No. If corresponding lengths are multiplied by a factor k, the base and perpendicular height both receive that factor. Their product, and therefore triangle area, is multiplied by k squared. You can verify this by calculating both areas directly.
Conclusion
Similar-triangle problems become manageable when you separate three decisions: why the triangles are similar, which parts correspond and what quantity the question requests. Use one consistent side ratio for lengths, substitute algebraic answers back into their expressions and calculate area with both dimensions scaled. Practise with triangles that face different directions. If the relationships remain clear after the picture changes, you are learning the method rather than memorizing the layout.