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Triangles: Similarity, BPT, and Pythagoras Without the Fog

This chapter becomes easy once you stop seeing triangles as static diagrams and start seeing them as scaled photos, ladder frames, kite strings, shadows, shortcuts, and smart ratio machines.

Similarity vs CongruenceBPT / ThalesAAA, SAS, SSSArea ScalingPythagorasConverse

Similarity vs. Congruence

Same shape does not always mean same size.

Congruent figures are identical in both shape and size. Similar figures have the same shape, but their sizes may differ.

A passport photograph and its tiny print on an ID card look alike in shape, but they are not equal in size. That is the idea of similarity.

Real-Life Hook

Taj Mahal and Its Souvenir Replica

The real Taj Mahal and a scaled souvenir model have the same shape, so they are similar.

Two identical souvenir pieces made from the same mould are congruent.

Formula Focus

\text{Similar} \Rightarrow \text{same shape, size may differ}

\text{Congruent} \Rightarrow \text{same shape and same size}

Memory trick

Congruent means carbon copy. Similar means same design, but scaled.

All circles

Every circle is similar to every other circle, because the shape stays the same even when the radius changes.

All squares

All squares are similar. Two squares are congruent only when their corresponding sides are equal.

Classify two circles

Two circles have radii 3 cm and 5 cm. Are they similar, congruent, or both?

Show solution

All circles are similar, so the pair is definitely similar.

The radii are different, so they are not congruent.

Therefore, they are similar but not congruent.

Practice

1. Two circles have radii 4 cm and 4 cm. Are they similar, congruent, or both?
2. Are all rectangles similar? Give one reason.
3. Two triangles have sides $(3,4,5)$ and $(6,8,10)$ . Are they similar?

Similar Polygons and Similar Triangles

Equal corresponding angles and proportional corresponding sides.

Two polygons are similar when their corresponding angles are equal and their corresponding sides are in the same ratio.

Triangles are especially important because several shorter tests are enough to prove similarity without checking everything one by one.

Correspondence Rule

The order of names matters. If $\triangle ABC\sim\triangle PQR$ , then $A\leftrightarrow P$ , $B\leftrightarrow Q$ , and $C\leftrightarrow R$ .

Formula Focus

\triangle ABC\sim\triangle DEF

\angle A=\angle D,\ \angle B=\angle E,\ \angle C=\angle F

\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}

Scale factor

If a small triangle grows to a larger similar triangle, the ratio of corresponding sides is called the scale factor.

Logo enlargement

A logo triangle with sides $3,4,5$ enlarged to $9,12,15$ keeps shape and changes only size, so it is similar.

Use proportional sides

In $\triangle ABC\sim\triangle DEF$ , if $AB=6$ , $BC=9$ , and $DE=10$ , find $EF$ .

Show solution

Since corresponding sides are proportional,

\frac{AB}{DE}=\frac{BC}{EF}

\frac{6}{10}=\frac{9}{EF}

So $6EF=90$ , hence $EF=15$ .

Practice

1. If the scale factor is 4, how do corresponding lengths change?
2. If one side pair is 7 and 14, what is the scale factor from smaller to larger?
3. If $AB:DE=2:5$ and $BC=8$ , find $EF$ .

Basic Proportionality Theorem

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

In a triangle, if a line is drawn parallel to one side, it cuts the other two sides proportionally. This is called the Basic Proportionality Theorem, or BPT.

It is also called Thales' theorem in many textbooks. The whole idea depends on the line being parallel to the third side.

Real-Life Hook

A-Frame Ladder

Imagine an A-frame ladder with a support bar fixed parallel to the ground. That support divides both slant legs in the same ratio.

Formula Focus

DE\parallel BC\Rightarrow \frac{AD}{DB}=\frac{AE}{EC}

What to look for

Whenever you see a line marked parallel to one side of a triangle, BPT should immediately come to mind.

Why parallel matters

Without parallelism, the two side divisions need not stay proportional.

Apply BPT directly

In $\triangle ABC$ , $DE\parallel BC$ , $AD=3$ , $DB=5$ , and $AE=6$ . Find $EC$ .

Show solution

By BPT,

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

\frac{3}{5}=\frac{6}{EC}

So $3EC=30$ , which gives $EC=10$ .

Practice

1. In $\triangle ABC$ , $DE\parallel BC$ , $AD=2$ , $DB=7$ , $AE=4$ . Find $EC$ .
2. If $AD:DB=5:3$ and $AE=15$ , find $EC$ .
3. Explain BPT using an A-frame ladder.

Converse of BPT

If a line divides two sides in the same ratio, then it is parallel to the third side.

The converse reverses BPT. Instead of starting with a parallel line, it starts with equal ratios and lets us prove parallelism.

This result is very common in board proofs because it turns a ratio statement into a geometry conclusion.

Proof Habit

When the question asks you to show a line is parallel, check whether you can form equal ratios on the two sides of the triangle.

Formula Focus

\frac{AD}{DB}=\frac{AE}{EC}\Rightarrow DE\parallel BC

Test for parallelism

In $\triangle ABC$ , $AD=3$ , $DB=6$ , $AE=4$ , and $EC=8$ . Is $DE\parallel BC$ ?

Show solution

Compute the two ratios:

\frac{AD}{DB}=\frac{3}{6}=\frac12

\frac{AE}{EC}=\frac{4}{8}=\frac12

The ratios are equal, so by the converse of BPT, $DE\parallel BC$ .

Practice

1. If $AD:DB=7:9$ and $AE:EC=7:9$ , what can you conclude?
2. Check whether $DE\parallel BC$ if $AD=2$ , $DB=3$ , $AE=6$ , $EC=8$ .
3. Create one numerical example where the converse fails.

Similarity Criteria: AAA, SAS, SSS

Three standard tests for proving that two triangles are similar.

There are three main similarity criteria for triangles: AAA, SAS, and SSS.

Each one gives us a shorter path to prove similarity without checking every angle and side separately.

Real-Life Hook

Shadows Under the Same Sun

A student and a lamppost standing under the same sunlight make triangles with equal angles. Their height and shadow triangles are similar by AAA.

Formula Focus

\text{AAA: all corresponding angles equal}

\text{SAS: included angle equal and including sides proportional}

\text{SSS: all corresponding sides proportional}

AAA

If all matching angles are equal, the triangles have the same shape.

SAS

This works only when the equal angle is the included angle between the proportional side pairs.

SSS

If all three side pairs are in one common ratio, the triangles are similar.

Use SSS similarity

Check whether the triangles with side sets $(3,4,5)$ and $(6,8,10)$ are similar.

Show solution

Compare corresponding sides:

\frac{3}{6}=\frac{4}{8}=\frac{5}{10}=\frac12

All three ratios are equal, so the triangles are similar by SSS.

Practice

1. Which criterion fits “same angles, different size”?
2. Check similarity for $(5,12,13)$ and $(10,24,26)$ .
3. A child 4 ft tall casts a 6 ft shadow. A pole casts a 24 ft shadow. Find the pole height.

Areas of Similar Triangles

When side lengths scale, areas scale by the square of that ratio.

If two triangles are similar and their corresponding sides are in the ratio $m:n$ , then their areas are in the ratio $m^2:n^2$ .

This is one of the most tested ideas in the chapter because students often forget to square the side ratio.

Common Mistake

Do not compare areas using the same ratio as the sides. Always square the side ratio first.

Real-Life Hook

The Big Samosa Rule

If a halwai doubles every side of a triangular samosa sheet, the size factor becomes 2 but the area factor becomes $2^2=4$ .

Formula Focus

\text{Side ratio }m:n\Rightarrow \text{Area ratio }m^2:n^2

Find an area ratio

The corresponding side ratio of two similar triangles is $2:3$ . Find the ratio of their areas.

Show solution

Area ratio is the square of side ratio.

So,

2^2:3^2=4:9

Hence the area ratio is $4:9$ .

Practice

1. If the area ratio is $25:49$ , find the side ratio.
2. A triangular logo is enlarged by factor 4. By what factor does the area change?
3. If side ratio is $7:9$ and smaller area is 49, find larger area.

Pythagoras Theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Pythagoras theorem applies only to right triangles. If the two legs are $a$ and $b$ , and the hypotenuse is $c$ , then $c^2=a^2+b^2$ .

This theorem appears in ladders, rectangular fields, kites, diagonals, and every problem where a right angle creates a direct shortcut.

Real-Life Hook

Kite String and Maidan Shortcut

A kite string, the height of the kite above the ground, and the horizontal distance below the kite form a right triangle.

The diagonal shortcut across a rectangular maidan is also the hypotenuse of a right triangle.

Formula Focus

c^2=a^2+b^2

Hypotenuse

The hypotenuse is always the side opposite the right angle, and it is the longest side.

When to use it

Use Pythagoras only when the triangle is right-angled or clearly treated as one.

Find the hypotenuse

Find the hypotenuse of a right triangle with legs 6 cm and 8 cm.

Show solution

Use Pythagoras theorem:

c^2=6^2+8^2=36+64=100

So $c=10$ cm.

Practice

1. Find the hypotenuse when the legs are 9 cm and 12 cm.
2. Find a leg if the hypotenuse is 17 cm and the other leg is 8 cm.
3. A rectangular park is 24 m by 7 m. Find the diagonal.

Converse of Pythagoras and Mixed Mastery

If the longest side satisfies the Pythagoras relation, the triangle is right-angled.

The converse of Pythagoras helps us test whether a triangle is right-angled from its side lengths alone.

If the square of the longest side equals the sum of the squares of the other two sides, then the triangle is right-angled.

Exam Survival Plan

Read the diagram before the numbers.
If you see parallel lines, think BPT or its converse.
If you see equal-angle triangles, think similarity.
If you see a right angle or longest side, think Pythagoras or its converse.

Formula Focus

c^2=a^2+b^2\Rightarrow \text{triangle is right-angled}

Check whether a triangle is right-angled

Check whether the sides 5 cm, 12 cm, and 13 cm form a right triangle.

Show solution

The longest side is 13.

13^2=169

5^2+12^2=25+144=169

Since both values are equal, the triangle is right-angled.

Practice

1. Check whether 8, 15, 17 form a right triangle.
2. Check whether 9, 10, 17 form a right triangle.
3. If one leg is 15 and the area is 90 sq units, find the other leg.

Triangles Summary

Similarity

Similar figures have the same shape. Congruent figures have the same shape and the same size.

BPT

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

Converse of BPT

If two sides are divided in the same ratio, the joining line is parallel to the third side.

AAA, SAS, SSS

These are the three main criteria used to prove two triangles are similar.

Area Rule

Area ratio of similar triangles equals the square of their corresponding side ratio.

Pythagoras

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

Converse of Pythagoras

If the longest side squared equals the sum of the squares of the other two sides, the triangle is right-angled.

Exam Lens

Parallel lines suggest BPT, equal-angle patterns suggest similarity, and longest-side checks suggest Pythagoras.

Practice and Revision

Use these chapter-level prompts to rehearse the main ideas before exams.

Quick Practice

Are all squares similar? Are all squares congruent?
If the side ratio of similar triangles is 3:4, what is the area ratio?
In BPT, if AD:DB = 2:5 and AE = 6, find EC.
Check whether the side sets (7,24,25) and (6,8,11) form right triangles.

Board-Style Thinking

A pole and its shadow form a triangle similar to a student and the student's shadow. Use this idea to find the pole height.
If areas of two similar triangles are 16 and 81, and one corresponding side of the smaller is 12 cm, find the matching side of the larger triangle.
If a line divides two sides of a triangle in the ratio 4:5 and 8:10, prove that it is parallel to the third side.

Open Practice

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