MathematicsClass 10ICSE

How a Shadow Lets You Measure a Tree Without Ever Climbing It

Learn the SSS, SAS and AA conditions for similar triangles, tell similarity from congruency, use the Basic Proportionality Theorem and its converse, and find unknown sides from the ratio of corresponding sides.

What does it mean for two triangles to be similar?

Two triangles are similar when they have the same shape but not necessarily the same size. Then:

- Corresponding angles are equal
- Corresponding sides are in the same ratio

We write

△ A B C \sim △ P QR

, with the letters in matching order, so

A

corresponds to

P

B

Q

and

C

R

.

This part covers the tests for similarity, the difference from congruency, the Basic Proportionality Theorem, and finding unknown sides.

What are the SSS, SAS and AA conditions for similarity, and how do you spot similar triangles in a figure?

Two triangles are similar if two angles of one equal two angles of the other (AA), if two sides are proportional and the included angles are equal (SAS), or if all three sides are proportional (SSS).

AA. If two angles match, the third must match too, since the angles add to

18 0^{\circ}

.

SAS. In

△ A B C

A B = 4

cm,

A C = 6

cm,

∠ A = 5 0^{\circ}

; in

△ P QR

P Q = 6

cm,

P R = 9

cm,

∠ P = 5 0^{\circ}

\frac{A B}{P Q} = \frac{4}{6} = \frac{2}{3} \frac{A C}{P R} = \frac{6}{9} = \frac{2}{3} so △ A B C \sim △ P QR

SSS. Sides

3, 4, 5

cm and

6, 8, 10

cm: every ratio is

\frac{1}{2}

, so the triangles are similar.

Spotting similarity in figures.

- A line parallel to one side cuts off a small triangle similar to the whole, by AA
- In a right triangle, the altitude to the hypotenuse creates two smaller triangles, each similar to the original

An everyday example. The two set squares in a geometry box often have the same angles but different sizes — similar triangles by AA.

The substance. The order of letters matters:

△ A B C \sim △ P QR

tells you exactly which sides correspond.

How is similarity different from congruency?

Congruent triangles have equal corresponding sides and angles, so they are identical in shape and size, while similar triangles have equal angles but only proportional sides, so they may differ in size.

Comparison:

- Angles — equal in both similar and congruent triangles
- Sides — equal in congruent triangles; in the same ratio in similar triangles
- Ratio of sides — exactly

1

for congruent triangles; any positive number for similar triangles
- Symbol —

≅

for congruent;

\sim

for similar

Worked example. Triangles with sides

5, 12, 13

cm and

10, 24, 26

cm:

\frac{5}{10} = \frac{12}{24} = \frac{13}{26} = \frac{1}{2}

They are similar but not congruent, because the ratio is not

1

.

An everyday example. A passport-size photo and an enlarged print of the same picture show exactly the same shapes at different sizes — similar, not congruent.

The substance. **Every pair of congruent triangles is similar, but similar triangles are congruent only when the ratio is

1

.** That is why three equal angles prove similarity but never congruence.

How do you use the Basic Proportionality Theorem and its converse to find unknown lengths?

If a line parallel to one side of a triangle cuts the other two sides, it divides them in the same ratio; conversely, if a line divides two sides in the same ratio, it is parallel to the third side.

In

△ A B C

with

D

A B

and

E

A C

D E ∥ B C ⟺ \frac{A D}{D B} = \frac{A E}{E C}

Worked example 1.

A D = 3

cm,

D B = 5

cm,

A E = 4.5

cm and

D E ∥ B C

. Find

E C

\frac{3}{5} = \frac{4.5}{E C} E C = 7.5 cm

Worked example 2 — converse.

A D = 2.4

D B = 3.6

A E = 3

E C = 4.5

cm.

\frac{2.4}{3.6} = \frac{2}{3} \frac{3}{4.5} = \frac{2}{3} so D E ∥ B C

Worked example 3 — algebra.

A D = x

D B = x - 2

A E = x + 2

E C = x - 1

and

D E ∥ B C

x (x - 1) = (x + 2) (x - 2) x^{2} - x = x^{2} - 4 x = 4

Check:

\frac{4}{2} = \frac{6}{3}

.

An everyday example. Horizontal bars on a triangular roof truss, parallel to the base, cut both sloping beams in the same ratio.

The trap. **

\frac{D E}{B C}

equals

\frac{A D}{A B}

, not

\frac{A D}{D B}

** — the parallel side compares with the whole side.

How do you calculate unknown sides using the ratio of corresponding sides of similar triangles?

Write the equal ratios of corresponding sides, find the scale factor from a known pair, and multiply or divide to get the unknown sides.

Worked example 1.

△ A B C \sim △ P QR

with

A B = 6

B C = 8

C A = 10

cm and

P Q = 9

cm.

k = \frac{P Q}{A B} = \frac{9}{6} = 1.5 QR = 8 \times 1.5 = 12 cm R P = 10 \times 1.5 = 15 cm

Worked example 2.

D E ∥ B C

A D = 3

cm,

D B = 2

cm,

D E = 6

cm. Find

B C

\frac{A D}{A B} = \frac{D E}{B C} \frac{3}{5} = \frac{6}{B C} B C = 10 cm

Worked example 3 — altitude in a right triangle. In

△ A B C

right-angled at

A

A D

is the altitude to

B C

, with

B D = 4

cm and

D C = 9

cm. The triangles

A B D

and

C A D

are similar, so

\frac{A D}{B D} = \frac{D C}{A D} A D^{2} = 4 \times 9 = 36 A D = 6 cm

An everyday example. **A

2

m pole casts a

3

m shadow while a tree nearby casts an

18

m shadow.** The sun's rays make the triangles similar:

\frac{h}{18} = \frac{2}{3} h = 12 m

The substance. Match sides by the equal angles opposite them, not by where they appear in the figure.

Exam tip

What earns full marks on similar triangles?

Name the criterion, list the matching angles or ratios, and write the similarity statement with letters in the correct order.

- For AA, give the two pairs of equal angles with reasons
- For SAS, show two ratios and the included angle
- For SSS, show all three ratios
- For BPT, state that the line is parallel before using the ratio
- **Write

△ A B C \sim △ P QR

with corresponding vertices in order
- Give units with every length

The trap.** Using

\frac{A D}{D B}

to find

D E

. **

D E

corresponds to

B C

, so use

\frac{A D}{A B}

.**

Did you know

How can a mirror on the ground help you find the height of a building?

Place a small mirror flat on the ground and step back until you see the top of a building in it.

Light reflects at equal angles, so the triangle formed by your eye, your feet and the mirror is similar to the triangle formed by the top of the building, its base and the mirror.

If your eyes are

1.5

m high, you stand

2

m from the mirror, and the mirror is

40

m from the building:

\frac{h}{40} = \frac{1.5}{2} h = 30 m

Exam relevance

Where does similarity reappear in JEE Main and NEET?

This is foundation work used in Class 11 Straight Lines in JEE Main and in Class 12 Ray Optics and Optical Instruments, part of both JEE Main and NEET Physics.

What gets built on. In Ray Optics, magnification is found from similar triangles formed by the object, the image and rays through the lens or mirror, giving the ratio of image height to object height. In coordinate geometry, the fact that slope is constant along a line rests on similar triangles, and the section formula can be proved with them.

Question types. Numericals in optics built on ratios of heights and distances, and coordinate problems using proportional division.

The trap that costs marks. Pairing the wrong sides because the triangles are drawn in different orientations.

Key takeaways

What must you be able to do from this part?

- Similar triangles: equal angles and proportional sides
- Criteria: AA, SAS with the included angle, SSS
- Congruent means ratio

1

; similar allows any ratio
- BPT:

D E ∥ B C

gives

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

; the converse proves parallel lines
- **

A D = 3

D B = 5

A E = 4.5

** gives

E C = 7.5

cm
- Unknown sides: scale factor from a known pair;

P Q = 9

gives

QR = 12

and

R P = 15

- Right triangle altitude:

A D^{2} = B D \times D C

On a sunny day, measure your own shadow and a lamp post's shadow, and work out the lamp post's height.

What does it mean for two triangles to be similar?

What are the SSS, SAS and AA conditions for similarity, and how do you spot similar triangles in a figure?

How is similarity different from congruency?

How do you use the Basic Proportionality Theorem and its converse to find unknown lengths?

How do you calculate unknown sides using the ratio of corresponding sides of similar triangles?

What earns full marks on similar triangles?

How can a mirror on the ground help you find the height of a building?

Where does similarity reappear in JEE Main and NEET?

What must you be able to do from this part?

Related study notes

Ready to put this into practice?