A right-angled triangle's two shorter sides are 7 and 6. Its longest side is √□.

85. The longest side squared = 7^2 + 6^2 = 85, so it is √85.

Shorter sides 7 and 7: the longest side squared = 7^2 + 7^2 = □ + □ = □. Fill in each blank

49, 49, 98. 7^2 = 49 and 7^2 = 49, which add to 98, so the longest side is √98.

To draw a line √10 long, what could a right-angled triangle's two shorter sides be? A. 1 and 10 B. 1 and 3 C. 1 and 9 D. 2 and 4 Choose the right answer

B. 1 and 3. 1^2 + 3^2 = 10, so shorter sides 1 and 3 give √10.

Surds & irrational numbers · Ages 13–14

Constructing lengths √n

Shorter sides a and b give a longest side a² + b².

How to do it

Method

Shorter sides a and b give a longest side √(a^2 + b^2)

2² + 3² = 4 + 9 = 13

Another way

The spiral, adding 1 each time

2 → 3 → 4 → 5 → 6 → 7

In full

An irrational n never ends as a decimal, but it can be drawn exactly. In a right-angled triangle the square of the longest side equals the sum of the squares of the other two (Pythagoras' theorem): with both shorter sides 1, the longest side squared is 1² + 1² = 2, so it is 2. To draw 13, find two squares that add to 13: 4 + 9 = 13, so shorter sides 2 and 3 give a longest side of 13. A number that isn't a sum of two squares, like 7, can be reached step by step: on the side 2 build another right-angled triangle with a side of 1, and its longest side is 2 + 1 = 3; then 3 and 1 give 4 = 2, then 5, and so on. Each new triangle adds 1 under the root: a square-root spiral. Carry a length drawn this way onto a number line with compasses, and n's place on the line is found.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
A right-angled triangle's two shorter sides are 7 and 6. Its longest side is .
Show the answer and working
Answer: 85
The longest side squared = 7² + 6² = 85, so it is 85.
Example 2
Shorter sides 7 and 7: the longest side squared = 7² + 7² = + = .
Fill in each blank
Show the answer and working
Answer: 49, 49, 98
7² = 49 and 7² = 49, which add to 98, so the longest side is 98.
Example 3
To draw a line 10 long, what could a right-angled triangle's two shorter sides be?
- A. 1 and 10
- B. 1 and 3
- C. 1 and 9
- D. 2 and 4
Choose the right answer
Show the answer and working
Answer: B. 1 and 3
1² + 3² = 10, so shorter sides 1 and 3 give 10.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.A right-angled triangle with shorter sides 6 and 3 has a longest side of 45.
True or false?
Answer:
2.A square-root spiral starts with shorter sides 1 and 1; each new right-angled triangle takes the last one's longest side and a side of 1 as its shorter sides. The 6th triangle's longest side is .
Answer:
3.Which length can be drawn as the longest side of one right-angled triangle whose shorter sides are whole numbers?
- A. 12
- B. 17
- C. 14
- D. 21
Choose the right answer
Answer:
4.A right-angled triangle with shorter sides 1 and has a longest side of 2.
Answer:
5.With shorter sides 5 and , the longest side is 21.
Answer:

All the answers

1. True
2. 7
3. B. 17
4. 1
5. 4

Free worksheet: Constructing lengths √nWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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