Pythagoras and Similar Shapes

Pythagoras’ Theorem #

The two shorter squares equal the longest.

In a right triangle the two shorter squares together equal the longest one

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Look for the right angle. This rule only works when one corner measures exactly 90°.

The side across from that corner is the longest. It is the hypotenuse.

Square the short sides: $3^2=9$3² = 9 and $4^2=16$4² = 16. Adding gives $25=5^2$25 = 5², so the hypotenuse is 5.

Name the sides instead of measuring them: $a^2+b^2=c^2$a² + b² = c², in every right triangle.

Proving Pythagoras by Dissection #

Cut the two squares up and they fill the third.

The same four triangles leave a hole of $c^2$c² one way and $a^2+b^2$a² + b² the other

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One right triangle: legs a and b, hypotenuse c. The claim: $a^2+b^2=c^2$a² + b² = c².

Four copies inside a square of side a + b. The hole between them is tilted: $c^2$c².

The triangle’s sharp angles add to 90°, so each corner of the hole is a true right angle.

Slide the same four triangles into two rectangles. The hole is now $a^2$a² and $b^2$b².

Same square, same four triangles — so this hole and the tilted one match: $a^2+b^2=c^2$a² + b² = c².

Finding a Shorter Side with Pythagoras’ Theorem #

Take the known square away instead of adding.

When the hypotenuse is known the missing square is a subtraction, not a sum

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This time the hypotenuse is known and one of the short sides is missing.

The two shorter squares still add to the longest, so subtract to find the missing one.

25 is what is left, and the side whose square is 25 is 5.

The Converse of Pythagoras’ Theorem #

Three lengths in, a right angle out.

If the two shorter squares add to the longest square then the triangle must be right angled

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Sides 3 and 4 meet at a hinged corner. At 90° the third side is 5: $3^2+4^2=5^2$3² + 4² = 5².

Close the hinge to 70° and the third side shortens to 4.1, which is under 5.

Open it to 110° and the third side lengthens to 5.8, which is over 5.

The third side grows steadily as the hinge opens, so it reaches 5 only at 90°.

Now three lengths can be tested: $8^2+15^2=289$8² + 15² = 289 and $17^2=289$17² = 289, so the corner is 90°.

Similar Shapes #

Equal angles, sides in the same ratio.

Similar shapes have equal angles and sides in the same ratio

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Two triangles can be the very same shape at completely different sizes.

Double every side: 4 becomes 8, and no angle changes.

Scale by 3 instead: every side triples and no angle changes. Such shapes are similar.

The sun’s rays are parallel, so a stick and a tree meet them at equal angles: the two triangles are similar.

The shadows give the scale factor, 6 ÷ 3 = 2, so the tree is the stick’s 2 m × 2 = 4 m tall.

Finding Similar Triangles in a Figure #

Spot the pair before any ratio is written.

Two triangles can hide in one figure — matching angles show you the similar pair

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A line parallel to the base cuts the triangle, making a smaller triangle at the top.

Both share the top angle and the corresponding angles are equal, so they are similar.

Bowtie triangles are similar too: the vertically opposite angles match, and so do the alternate angles.

Corresponding sides face the matching angles: 2 against 4 gives a scale factor of 2.