Similar Triangles

What this move does

If you can translate, rotate and reflect a triangle, and then enlarge or shrink it, so that it fits exactly on another one, the two triangles are similar. The scale factor $k$ is called the ratio of similarity. Congruence is similarity with $k = 1$ .

△ A B C \sim △ A^{'} B^{'} C^{'} ⟺ ⎩ ⎨ ⎧ \frac{A ^{'} B ^{'}}{A B} = \frac{B ^{'} C ^{'}}{B C} = \frac{C ^{'} A ^{'}}{C A} = k ∠ A = ∠ A^{'}, ∠ B = ∠ B^{'}, ∠ C = ∠ C^{'}

Proving similarity takes less than proving congruence: scaling does not change angles, so two pairs of equal angles are enough (AA). There are also “two sides in proportion and the included angles equal” (SAS) and “all three sides in proportion” (SSS).

Solving with similar triangles takes three steps:

Find a pair of triangles with the same shape. The most common sources: parallel lines (the “A shape” and the “hourglass”), a shared angle, a right angle plus one more equal angle.
Line up the corresponding vertices in the similarity statement, then write the proportions by position.
Algebra: in a proportion, knowing three quantities gives you the fourth; for areas, use $k^{2}$ .

Predict first

Every side of a triangle is enlarged to $3$ times its length. How many times bigger is the area?

3

times

6

times

9

times

27

times

Choose an answer first; then you can see the result.

Work through one

Move △ABC onto △A′B′C′: one extra step, scaling

Step 1 / 4

Compare the sizes
Both triangles go the same way round, so no reflection is needed. But the dashed triangle is clearly bigger: with translation and rotation alone, the corresponding vertices will never line up.
Translate
Rotate
Scale

The most common source: parallel lines

In $△ A B C$ draw a line $D E$ parallel to $B C$ . Then $∠ A D E = ∠ A B C$ (corresponding angles) and $∠ A$ is shared, so $△ A D E \sim △ A B C$ .

Drag

D

. Of the four ratios, which three are always equal? How is the fourth related to them?

Example: DE ∥ BC, AD = 4, DB = 6, BC = 15. Find DE

Step 1 / 4

Find the similar triangles
$D E ∥ B C$ , so $△ A D E \sim △ A B C$ (the small orange triangle and the whole big triangle).
Which sides correspond?
Write the proportion
The area, while we are here

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: $\frac{A D}{A B} = \frac{A E}{A C} = \frac{D E}{B C}$

$E$ follows $D$ , so $D E$ always stays parallel to $B C$ .

Practice

Exercise 1Basic

$△ A B C \sim △ D E F$ with ratio of similarity $A B : D E = 2 : 3$ . The area of $△ A B C$ is $12$ . Find the area of $△ D E F$ .

Exercise 2Basic

In $△ A B C$ , $D$ lies on $A B$ and $E$ on $A C$ with $D E ∥ B C$ . Given $A D = 3$ , $D B = 5$ and $A C = 20$ , find the length of $A E$ .

Exercise 3Advanced

In trapezoid $A B C D$ , $A B ∥ C D$ , $A B = 6$ , $C D = 10$ , and the diagonals $A C$ and $B D$ meet at $O$ . The area of $△ A O B$ is $9$ . Find the area of $△ C O D$ .

Exercise 4Challenge

In $△ A B C$ , $∠ A C B = 9 0^{\circ}$ and $C D ⊥ A B$ at $D$ . Given $A D = 4$ and $D B = 9$ , find the length of $C D$ .

What this move does

Work through one

Move △ABC onto △A′B′C′: one extra step, scaling

The most common source: parallel lines

Example: DE ∥ BC, AD = 4, DB = 6, BC = 15. Find DE

When it fails

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Using the area ratio as the ratio of similarity

Wrong denominator in a ratio cut by a parallel line