Similar triangles

Q: Are all equilateral triangles similar?

Yes. All their angles are 60^\circ, so any two are similar by AA.

Two triangles are similar when they have the same shape: their angles are equal and their sides are in the same ratio. Similar triangles let you find lengths you cannot measure directly, such as the height of a tower from its shadow.

Formulas

If ΔABC ~ ΔPQR

\frac{A B}{PQ} = \frac{BC}{QR} = \frac{C A}{RP}

Basic proportionality theorem (DE ∥ BC)

\frac{A D}{D B} = \frac{A E}{EC}

Areas of similar triangles

\frac{area _{1}}{area _{2}} = (\frac{side _{1}}{side _{2}})^{2}

Worked examples

Example 1. In triangle ABC, DE ∥ BC with AD = 3 cm, DB = 6 cm and AE = 4 cm. Find EC.

$\frac{A D}{D B} = \frac{A E}{EC}, so \frac{3}{6} = \frac{4}{EC}$ .
$EC = \frac{4 \times 6}{3} = 8$ .

Answer: 8 cm

Example 2. ΔABC ~ ΔPQR with AB = 4 cm, PQ = 10 cm and BC = 6 cm. Find QR.

$\frac{A B}{PQ} = \frac{BC}{QR}, so \frac{4}{10} = \frac{6}{QR}$ .
$QR = \frac{6 \times 10}{4} = 15$ .

Answer: 15 cm

Example 3. A 6 m pole casts a 4 m shadow. At the same time a tower casts a 28 m shadow. How tall is the tower?

The sun's rays make similar triangles, so $\frac{height}{shadow}$ is the same.
$\frac{h}{28} = \frac{6}{4}, so h = 42$ .

Answer: 42 m

Example 4. Two similar triangles have sides in the ratio 2 : 3. The smaller has area

32 cm^{2}

. Find the area of the larger.

Areas are in the ratio $2^{2} : 3^{2} = 4 : 9$ .
$32 \times \frac{9}{4} = 72$ .

Answer:

72 cm^{2}

Common mistakes

Matching the wrong sides. Pair sides in the order the vertices are written: in ΔABC ~ ΔPQR, AB goes with PQ.
Using the side ratio for areas. Areas go with the SQUARE of the side ratio.
Confusing similar with congruent. Congruent triangles are the same size as well; similar ones only need the same shape.

Practice questions with answers

1. Two triangles are similar if their corresponding angles are equal and sides are in ____.

ASame difference
BSame ratio (proportion)
CSame sum
DSame product

Show answer

Answer: B) Same ratio (proportion)

Definition of similar triangles: angles equal AND sides proportional.

2. Two triangles have angles 30°, 60°, and 90°. Are they similar?

AYes, by AA criterion
BNo, angles are not enough
CYes, by SSS criterion
DNo, they must be congruent

Show answer

Answer: A) Yes, by AA criterion

Triangles with two equal angles are similar by the AA criterion.

3. The Basic Proportionality Theorem (Thales' Theorem) states that if a line is drawn parallel to one side of a triangle, then it divides the other two sides in the same _____.

Aratio
Bangle
Carea
Dlength

Show answer

Answer: A) ratio

BPT states the line parallel to one side divides the other two sides in the same ratio.

4. Which criterion proves two triangles similar if two angles of one equal two angles of the other?

AAA criterion
BSSS criterion
CSAS criterion
DRHS criterion

Show answer

Answer: A) AA criterion

If two pairs of corresponding angles are equal, the triangles are similar by the AA (Angle-Angle) criterion.

5. The ratio of areas of two similar triangles equals the ratio of the squares of their corresponding ____.

Asides
Bangles
Cperimeters cubed
Dheights added

Show answer

Answer: A) sides

By the Area Theorem, area ratio =

(r a t i oo f corres p o n d in g s i d es)^{2}

, so it equals the ratio of squares of corresponding sides.

6. Two circles are always similar to each other.

ATrue
BFalse
COnly if equal radius
DOnly if concentric

Show answer

Answer: A) True

Any two circles have the same shape and one is a scaled copy of the other, so all circles are similar.

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Frequently asked questions

How do you prove two triangles are similar?

Show two pairs of equal angles (AA), all three sides in the same ratio (SSS), or two sides in the same ratio with the angle between them equal (SAS).

What is the basic proportionality theorem?

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

Are all equilateral triangles similar?

Yes. All their angles are $6 0^{\circ}$ , so any two are similar by AA.

Is congruence a special case of similarity?

Yes. Congruent triangles are similar triangles whose sides are in the ratio 1 : 1.

Similar triangles

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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