Pythagoras

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Lesson notes

Pythagoras' Theorem

Pythagoras was a Greek mathematician who lived over 2500 years ago.
Pythagoras' theorem links the lengths of the three sides of a right-angled triangle.
The hypotenuse is the longest side, opposite the right angle.
Formula: a² + b² = c², where c is the hypotenuse and a, b are the shorter sides.
It does not matter which shorter side is labelled a or b.

Right-angled triangle

Finding the Hypotenuse

To find the hypotenuse: square the two shorter sides, add them, then take the positive square root.
Formula: c = √(a² + b²).
When finding the hypotenuse, you add inside the square root.
Example: If a = 3, b = 4, then c = √(3² + 4²) = √25 = 5.

Finding a Shorter Side

To find a shorter side: square the hypotenuse and the other shorter side, subtract the smaller from the larger, then take the square root.
Formula: a = √(c² – b²).
When finding a shorter side, you subtract inside the square root.
Ensure the hypotenuse is longer than any other side; otherwise, check your working.

Using Pythagoras with Other Shapes

Pythagoras can be applied to any shape that can be split into right-angled triangles.
For a rectangle or square, the diagonal splits the shape into two identical right-angled triangles.
The diagonal becomes the hypotenuse of those triangles.
Example: In a rectangle 14 cm by w cm with diagonal 23 cm, use Pythagoras to find w: w = √(23² – 14²).

Multi-Step Problems

In multi-step problems, leave intermediate answers as exact values (e.g., √63) to avoid rounding errors.
Only round the final answer to the required degree of accuracy.
Work through each right-angled triangle step by step.
Example: Find BC in a trapezium by first finding BD using triangle ABD, then DC = AC – AD, then BC = √(BD² + DC²).

Worked example: two right-angled triangles

Common Mistakes & Tips

The hypotenuse must be the longest side; if it isn't, you've made an error.
When subtracting for a shorter side, always subtract the smaller square from the larger square.
Otherwise, you'll get a negative number and a 'Math Error' when taking the square root.
Always check that your answer is reasonable.

Slides

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Practice questions

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1.Which side of a right-angled triangle is the hypotenuse?
Easy
- AThe side opposite the right angle
- BThe side adjacent to the right angle
- CThe shortest side
- DAny side
2.Pythagoras' theorem states that for a right-angled triangle with sides a, b and hypotenuse c:
Easy
- Aa² + b² = c²
- Ba + b = c
- Ca² = b² + c²
- Da² - b² = c²
3.A right-angled triangle has legs of lengths 3 cm and 4 cm. What is the length of the hypotenuse?
Easy
- A5 cm
- B7 cm
- C12 cm
- D25 cm
4.A flagpole is 25 m tall. A rope from the top is tied to the ground 8 m from the base. What is the length of the rope?
Medium
- A26.2 m
- B24.0 m
- C33.0 m
- D17.0 m
5.A rectangle is 14 cm wide and has a diagonal of 23 cm. What is the length of the rectangle?
Medium
- A18.2 cm
- B26.9 cm
- C9.0 cm
- D18.0 cm
6.A right-angled triangle has legs of lengths x and 8 cm, and hypotenuse 17 cm. What is x?
Easy
- A15 cm
- B18.8 cm
- C9 cm
- D25 cm
7.The length of one side of a rectangle is 12 cm and the diagonal is 13 cm. What is the area of the rectangle?
Medium
- A60 cm²
- B156 cm²
- C30 cm²
- D144 cm²
8.A trapezium has AB = 150 m, BC = 90 m, CD = 120 m, with angle ABC = angle BCD = 90°. What is the length AD, correct to the nearest metre?
Hard
- A95 m
- B85 m
- C108 m
- D122 m

Revision notes

## Pythagoras Theorem - **Pythagoras** was a Greek mathematician (2500+ years ago). - **Pythagoras' theorem** links the three sides of a **right-angled triangle**. - The **hypotenuse** is the longest side, opposite the right angle. - Formula: **a² + b² = c²** where c is the hypotenuse, a and b are the shorter sides. ### Finding the Hypotenuse - Square the two shorter sides, add them, then take the positive square root. - c = √(a² + b²) ### Finding a Shorter Side - Square the hypotenuse and the other shorter side, subtract, then take the positive square root. - a = √(c² – b²) ### Tips - The hypotenuse must be the longest side. If it's not, check your working. - When finding a shorter side, subtract the smaller square from the larger to avoid a negative under the square root. - In multi-step problems, keep answers exact (e.g., √63) until the final step, then round. - Pythagoras can be used with any shape that can be split into right-angled triangles (e.g., rectangles, squares). ### Common Question Types - Find the hypotenuse given two shorter sides. - Find a shorter side given the hypotenuse and one side. - Find the diagonal of a rectangle or square. - Combine with other shapes (trapezium, multiple triangles). - Calculate perimeter or area after finding missing lengths. > **Example**: In a right triangle with legs 12 cm and 9 cm, hypotenuse = √(12²+9²) = √(144+81) = √225 = 15 cm.

Pythagoras' theorem: a² + b² = c², where c is the length of the hypotenuse and a, b are the lengths of the two shorter sides of a right-angled triangle.
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Right-angled triangle: A triangle with one angle equal to 90°.

Flashcards

What is Pythagoras' theorem?

For a right-angled triangle, a² + b² = c², where c is the hypotenuse (longest side) and a, b are the two shorter sides.

How do you find the length of the hypotenuse using Pythagoras?

Square the two shorter sides, add them, then take the positive square root: c = √(a² + b²).

How do you find the length of a shorter side using Pythagoras?

Square the hypotenuse and the other shorter side, subtract the smaller square from the larger, then take the positive square root: a = √(c² - b²).

What is the hypotenuse?

The longest side of a right-angled triangle, opposite the right angle.

In a right triangle with legs 3 cm and 4 cm, find the hypotenuse.

c = √(3² + 4²) = √(9+16) = √25 = 5 cm.

In a right triangle with hypotenuse 13 cm and one leg 5 cm, find the other leg.

a = √(13² - 5²) = √(169-25) = √144 = 12 cm.

A flagpole is 25 m tall. A rope from its top is tied to the ground 8 m from its base. Find the rope length.

Rope length = √(25² + 8²) = √(625+64) = √689 ≈ 26.2 m.

A rectangle has sides 14 cm and w cm, diagonal 23 cm. Find w.

w = √(23² - 14²) = √(529-196) = √333 ≈ 18.2 cm.

Find the diagonal of a rectangle with sides 12 cm and 5 cm.

Diagonal = √(12² + 5²) = √(144+25) = √169 = 13 cm.

A rectangle has one side 12 cm and diagonal 13 cm. Find its area.

Other side = √(13² - 12²) = √(169-144) = √25 = 5 cm. Area = 12 × 5 = 60 cm².

A guy rope of length 6.9 m is attached to a flagpole top and fixed 3.2 m from its base. Find the flagpole height.

Height = √(6.9² - 3.2²) = √(47.61 - 10.24) = √37.37 ≈ 6.11 m.

In triangle ABC, AB = 12 cm, AD = 9 cm, AC = 22 cm, and BD is perpendicular to AC. Find BC (x).

BD = √(12² - 9²) = √63 ≈ 7.937 cm. DC = 22 - 9 = 13 cm. BC = √(BD² + DC²) = √(63+169) = √232 ≈ 15.2 cm.

A trapezium has AB=150 m, BC=90 m, CD=120 m, with right angles at B and C. Show AD ≈ 95 m.

Drop perpendicular from D to AB, forming a right triangle with legs (150-120)=30 m and 90 m. AD = √(30² + 90²) = √(900+8100) = √9000 ≈ 94.9 m, correct to nearest metre 95 m.

For the trapezium above, find the cost of a fence at $48 per 5-metre section.

Perimeter = AB+BC+CD+AD = 150+90+120+95 = 455 m. Number of sections = 455/5 = 91. Cost = 91 × $48 = $4368.

A right triangle has sides 5 cm, 12 cm, and 13 cm. Find its perimeter.

Perimeter = 5 + 12 + 13 = 30 cm.

A rectangular pond ABCD has AB = 12 m, diagonal AC = 15 m. Find BC.

BC = √(15² - 12²) = √(225-144) = √81 = 9 m.

For the pond above, find the length of fence around its edge.

Perimeter = 2(12+9) = 42 m.

What common mistake should you avoid when using Pythagoras?

Ensure the hypotenuse is the longest side; if it ends up shorter than another side, you made an error. When finding a shorter side, subtract the smaller square from the larger to avoid a negative under the square root.

How can Pythagoras be used with rectangles?

The diagonal splits the rectangle into two right-angled triangles, with the diagonal as the hypotenuse. So diagonal² = length² + width².

In a right triangle, if a = 6 and b = 8, find c.

c = √(6² + 8²) = √(36+64) = √100 = 10.

In a right triangle, if c = 17 and a = 15, find b.

b = √(17² - 15²) = √(289-225) = √64 = 8.

A 25 m flagpole has a rope from top to ground 8 m from base. Find rope length (exact form).

Rope = √(25² + 8²) = √(625+64) = √689 m.

A rectangle 14 cm by w cm has diagonal 23 cm. Find w (exact form).

w = √(23² - 14²) = √(529-196) = √333 cm.

A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.

c = √(9² + 12²) = √(81+144) = √225 = 15 cm.

A right triangle has hypotenuse 10 cm and one leg 6 cm. Find the other leg.

Other leg = √(10² - 6²) = √(100-36) = √64 = 8 cm.

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