SPI Mathematics · Triangles

Similar Triangles and Proportional Sides

Similar triangles have the same shape: equal angles and every side multiplied by the same number, and a line parallel to one side, or a line bisecting an angle, cuts the other sides in a fixed ratio.

All Triangles notes

Why this matters

Eleven questions in the bank: seven EASY, two MODERATE and the chapter's two HARD ones. Nearly all come down to one proportion. The marks go when the wrong sides are paired, so the order of letters from the congruence page matters even more here.

Concept 1 of 4: When two triangles are similar

A photo and its enlargement show the same shape at different sizes. Triangles are the same: if the angles match, the sides all grow by one scale factor, and if the sides grow by one factor, the angles match.

Definition

Two triangles are similar (written $\sim$ ) when one of these holds:

AA (or AAA): two pairs of angles equal. The third pair is then equal too.
SSS: all three pairs of sides in the same ratio.
SAS: two pairs of sides in the same ratio and the angles between them equal.

As with congruence, $\triangle ABC \sim \triangle DEF$ pairs A with D, B with E and C with F.

Criterion	What you must show	Example
AA	Two pairs of equal angles	Angles 40°, 60° in one and 40°, 60° in the other
SSS	Three pairs of sides in one ratio	Sides 3, 4, 5 and 6, 8, 10 (ratio 1 : 2)
SAS	Two pairs of sides in one ratio, included angles equal	4 and 6 with 50° between; 6 and 9 with 50° between
Parallel line	A line parallel to one side cuts off a small triangle	Gives equal corresponding angles, so AA
Every pair of equilateral triangles	Nothing more: all angles are 60°	Always similar Congruent triangles are similar too, with ratio 1 : 1.

Similar means the same shape; congruent means the same shape and the same size.

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The same idea in a practice question:

Practice questionModerate

Example 1 · Triangles · Similar Triangles and Proportional Sides

\triangle

ABC is similar

\triangle

ADE then the measure of angle D =

Finding the angle at the wrong corner

\triangle ABC \sim \triangle ADE

, the angle at D matches the angle at B because both are in second place. Find the angle at B first, then copy it to D.

Drill 1 more on when two triangles are similar

Concept 2 of 4: Corresponding sides are in one ratio

Once two triangles are similar, one is a scaled copy of the other. Find the scale factor from one known pair of matching sides, then multiply any other side by it.

Definition

In $\triangle ABC \sim \triangle DEF$ : $\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{CA}{FD}$ .
Perimeters, heights and medians grow by the same factor as the sides.
Shadows: at the same moment the sun's rays make the same angle everywhere, so height over shadow is the same for every object.
Two lines crossing between two parallel lines make two similar triangles, joined at the crossing point.

Ratio of corresponding sides

\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \qquad \frac{\text{height}_1}{\text{shadow}_1} = \frac{\text{height}_2}{\text{shadow}_2}

Worked example

\triangle ABC \sim \triangle DEF

with AB = 4 cm, BC = 6 cm and DE = 10 cm. Find EF.

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The same idea in a practice question:

Practice questionModerate

Example 2 · Triangles · Similar Triangles and Proportional Sides

In the given figure,

\Delta ACB \sim \Delta APQ

. If AB = 6 cm, BC = 8 cm, and PQ = 4 cm then AQ is equal to

Pairing sides by their position on the page

\triangle ACB \sim \triangle APQ

, C matches P and B matches Q, so CB pairs with PQ and AB pairs with AQ. Read the pairs from the statement, never from how the figure looks.

Turning the ratio upside down

If the second triangle is the bigger one, its sides are the small sides TIMES the factor. Check the answer is bigger when it should be: a 2 m stick with a 3 m shadow means the tower is shorter than its shadow.

Drill 4 more on corresponding sides are in one ratio

Concept 3 of 4: Basic proportionality theorem

Draw a line across a triangle parallel to its base. It cuts off a smaller triangle of the same shape, so it divides the two slanting sides in the same ratio.

Definition

If DE is parallel to BC, with D on AB and E on AC, then $\dfrac{AD}{DB} = \dfrac{AE}{EC}$ .
Also $\dfrac{AD}{AB} = \dfrac{AE}{AC}$ : part over whole on both sides.
Converse: if a line divides two sides in the same ratio, it is parallel to the third side.
Dividing a segment AB in the ratio $m : n$ : draw a ray AX, mark $m + n$ equal steps $A_1, A_2, \ldots$ on it, join B to the last one, $A_{m+n}$ , and draw a line through $A_m$ parallel to it.

Basic proportionality theorem

DE \parallel BC \Rightarrow \frac{AD}{DB} = \frac{AE}{EC} \quad \text{and} \quad \frac{AD}{AB} = \frac{AE}{AC}

Worked example

\triangle ABC

, DE is parallel to BC with D on AB and E on AC. AD = 3 cm, DB = 4 cm and AE = 4.5 cm. Find EC.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Triangles · Similar Triangles and Proportional Sides

In the given figure line

l

is parallel to the side TR then length of SP =

x

is equal to

Mixing a part with a whole

\frac{AD}{DB} = \frac{AE}{EC}

compares part with part;

\frac{AD}{AB} = \frac{AE}{AC}

compares part with whole.

\frac{AD}{AB} = \frac{AE}{EC}

mixes the two and is wrong.

Joining B to the wrong point

To divide AB in the ratio 4 : 7, B is joined to

A_{11}

, the last of

4 + 7

steps. The parallel line then goes through

A_4

Drill 1 more on basic proportionality theorem

Concept 4 of 4: The angle bisector divides the opposite side

A line that cuts an angle of a triangle in half meets the opposite side closer to the shorter arm. It splits that side in the same ratio as the two sides that make the angle.

Definition

If AD bisects $\angle A$ of $\triangle ABC$ , with D on BC, then $\dfrac{BD}{DC} = \dfrac{AB}{AC}$ .
Each part of BC goes with the side it touches: BD with AB, DC with AC.
If AB = AC, the bisector meets BC at its midpoint.

Angle bisector theorem

\frac{BD}{DC} = \frac{AB}{AC}

Worked example

\triangle PQR

, PS bisects

\angle P

and meets QR at S. PQ = 6 cm, PR = 9 cm and QR = 10 cm. Find QS and SR.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Triangles · Similar Triangles and Proportional Sides

\triangle

ABC, AD is the bisector of

\angle

A AB = 3.4 cm BD = 4 cm BC = 10 cm then AC =

Pairing a part with the far side

BD sits next to AB, so

\frac{BD}{DC} = \frac{AB}{AC}

. Writing

\frac{BD}{DC} = \frac{AC}{AB}

puts the longer part next to the shorter side.

Drill 1 more on the angle bisector divides the opposite side

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Corresponding sides are in one ratio
Ratio of corresponding sides
$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \qquad \frac{\text{height}_1}{\text{shadow}_1} = \frac{\text{height}_2}{\text{shadow}_2}$
Basic proportionality theorem
Basic proportionality theorem
$DE \parallel BC \Rightarrow \frac{AD}{DB} = \frac{AE}{EC} \quad \text{and} \quad \frac{AD}{AB} = \frac{AE}{AC}$
The angle bisector divides the opposite side
Angle bisector theorem
$\frac{BD}{DC} = \frac{AB}{AC}$

Reference tables (1)

When two triangles are similar5 rows

Criterion	What you must show	Example
AA	Two pairs of equal angles	Angles 40°, 60° in one and 40°, 60° in the other
SSS	Three pairs of sides in one ratio	Sides 3, 4, 5 and 6, 8, 10 (ratio 1 : 2)
SAS	Two pairs of sides in one ratio, included angles equal	4 and 6 with 50° between; 6 and 9 with 50° between
Parallel line	A line parallel to one side cuts off a small triangle	Gives equal corresponding angles, so AA
Every pair of equilateral triangles	Nothing more: all angles are 60°	Always similar Congruent triangles are similar too, with ratio 1 : 1.

Similar means the same shape; congruent means the same shape and the same size.

Watch out for (6)

Finding the angle at the wrong corner
→ When two triangles are similar
Pairing sides by their position on the page
→ Corresponding sides are in one ratio
Turning the ratio upside down
→ Corresponding sides are in one ratio
Mixing a part with a whole
→ Basic proportionality theorem
Joining B to the wrong point
→ Basic proportionality theorem
Pairing a part with the far side
→ The angle bisector divides the opposite side

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