Pythagoras flashcards
10 cards on Pythagoras: the key points, worked examples and the slips to watch. Swipe through them, or print them.
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Pythagoras
Pythagoras' theorem
What does Pythagoras' theorem say, and when can you use it?
Answer it in your head, then turn the card over.
Pythagoras' theorem
Key points
- In a right-angled triangle, the two shorter sides squared and added make the hypotenuse squared.
- The hypotenuse is opposite the right angle, and is always the longest side.
- It links the three sides and needs no angles.
- No right angle, no Pythagoras: a triangle without one needs the cosine rule.
Formula
a² + b² = c², where c is the hypotenuse
Sides of 3, 4 and 5
3² + 4² = 9 + 16 = 25
5² = 25, so the sides fit a² + b² = c²
Watch out
When the triangle is turned round, find the right angle first: the side opposite it is c.
Pythagoras
Finding the hypotenuse
How do you find the length of the hypotenuse?
Answer it in your head, then turn the card over.
Finding the hypotenuse
Key points
- Square the two shorter sides and add them: that is c².
- Then square-root to get the length.
- Add first, then root: = = 17, not 8 + 15.
- Check: the hypotenuse is longer than either side, but shorter than the two added.
Example
Sides 5 cm and 12 cm: = = 13 cm
Sides 7 cm and 9 cm: = = 11.4 cm (1 dp)
Watch out
Forgetting the square root: sides of 6 and 8 give a hypotenuse of 10, not 100.
Pythagoras
Pythagorean triples
Which whole-number sides fit Pythagoras' theorem, and how can they save time?
Answer it in your head, then turn the card over.
Pythagorean triples
Key points
- 3, 4, 5 and its multiples: 6, 8, 10 and 9, 12, 15.
- 5, 12, 13 and 8, 15, 17 are the next ones to know.
- The largest number is always the hypotenuse.
- Spotting one gives the answer without a calculator.
Shorter sides of 9 and 12
9, 12 is 3, 4 multiplied by 3
So the hypotenuse is 5 × 3 = 15
Check: = = 15
Watch out
Shorter sides of 5 and 13 are not a 5, 12, 13 triangle: there, 13 is the hypotenuse.
Pythagoras
Finding a shorter side
How do you find one of the shorter sides when you know the hypotenuse?
Answer it in your head, then turn the card over.
Finding a shorter side
Key points
- Square the hypotenuse and the side you know.
- Subtract: hypotenuse² − the known side².
- Then square-root to get the length.
Formula
b² = c² − a²
Hypotenuse 13 cm, one side 5 cm
b² = 169 − 25 = 144
b = = 12 cm
Watch out
Adding when the hypotenuse is known gives a side longer than the hypotenuse, which is impossible.
Pythagoras
Pythagoras in a problem
How do you use Pythagoras' theorem in a worded problem, such as a ladder against a wall?
Answer it in your head, then turn the card over.
Pythagoras in a problem
Key points
- Sketch the right-angled triangle and mark the right angle.
- Label the hypotenuse: for a ladder against a wall, it is the ladder.
- Decide whether to add (finding the hypotenuse) or subtract (finding a shorter side).
- Give the units, and round only at the end.
A 5 m ladder, foot 1.5 m from the wall
Height² = 5² − 1.5² = 25 − 2.25 = 22.75
Height = = 4.77 m (2 dp)
Watch out
The ladder is the hypotenuse, so subtract: adding gives a wall height longer than the ladder.
Pythagoras
The height of an isosceles triangle
How do you find the height of an isosceles triangle when you know its sides?
Answer it in your head, then turn the card over.
The height of an isosceles triangle
Key points
- Draw the height from the top to the middle of the base.
- It splits the triangle into two right-angled triangles.
- Each has the equal side as its hypotenuse and half the base as a shorter side.
Sides 10 cm, 10 cm and base 12 cm
Half the base = 6 cm
Height = = = 8 cm
Area = × 12 × 8 = 48 cm²
Watch out
Use half the base, not all of it: the height meets the base at its midpoint.
Pythagoras
Testing for a right angle
How do you decide whether a triangle is right-angled from its three sides?
Answer it in your head, then turn the card over.
Testing for a right angle
Key points
- Square the two shorter sides and add them.
- Square the longest side.
- If the two answers are equal, the triangle has a right angle; if not, it does not.
- Show both numbers, then say what they prove in words.
Example
20, 21, 29: 400 + 441 = 841 and 29² = 841, so it is right-angled
6, 8, 11: 36 + 64 = 100 but 11² = 121, so it is not
Watch out
Always test the two shorter sides against the longest one, never the other way round.
Pythagoras
Distance between two points
How do you find the distance between two points on a coordinate grid?
Answer it in your head, then turn the card over.
Distance between two points
Key points
- Find the horizontal gap (the difference in x) and the vertical gap (the difference in y).
- They are the shorter sides of a right-angled triangle; the distance is its hypotenuse.
- Square both gaps, add, then square-root.
From (1, 2) to (7, 10)
Gaps: 7 − 1 = 6 across and 10 − 2 = 8 up
Distance = = = 10
Watch out
Adding the gaps, 6 + 8 = 14, gives the route along the grid lines, not the straight distance.
Pythagoras
HigherThe diagonal of a cuboid
How do you find the longest diagonal of a cuboid?
Answer it in your head, then turn the card over.
The diagonal of a cuboid
Key points
- Use Pythagoras twice.
- First across the base: its length and width give the base diagonal.
- Then the base diagonal and the height give the space diagonal.
- Sketch each right-angled triangle flat before you calculate.
Formula
d =
A 3 cm by 4 cm by 12 cm box
Base diagonal = = = 5 cm
Space diagonal = = = 13 cm
Watch out
The second triangle uses the base diagonal, not an edge of the box.
Pythagoras
HigherThe height of a pyramid
How do you find the vertical height of a square-based pyramid from its edges?
Answer it in your head, then turn the card over.
The height of a pyramid
Key points
- The height runs from the top straight down to the centre of the base.
- First find the distance from the centre of the base to a corner: half the base diagonal.
- The slanting edge is the hypotenuse, so subtract to find the height.
Base 6 cm by 6 cm, slanting edges 5 cm
Centre to corner: 3² + 3² = 18, so it is
Height² = 5² − 18 = 7
Height = = 2.65 cm (2 dp)
Watch out
Centre to corner is half the diagonal, , not half the side, 3 cm.
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Put it into practice
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