Numbers and operations · Ages 13–14

Constructing lengths √n worksheet

Part of: Surds & irrational numbers

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Numbers and operations · Ages 13–14

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.A right-angled triangle's two shorter sides are 3 and 3. Its longest side is .
Answer:
2.Shorter sides 8 and 6: the longest side squared = 8² + 6² = + = .
Fill in each blank
Answer:
3.A right-angled triangle's two shorter sides are 4 and 5. Its longest side is .
Answer:
4.A right-angled triangle with shorter sides 1 and has a longest side of 37.
Answer:
5.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 10th triangle's longest side is .
Answer:
6.A right-angled triangle with shorter sides 4 and has a longest side of 25.
Answer:
7.With shorter sides 5 and , the longest side is 14.
Answer:

8.To draw a line 20 long, what could a right-angled triangle's two shorter sides be?
- A. 2 and 18
- B. 3 and 5
- C. 1 and 20
- D. 2 and 4
Choose the right answer
Answer:
9.A right-angled triangle with shorter sides 4 and 4 has a longest side of 32.
True or false?
Answer:
10.A right-angled triangle with shorter sides 5 and 2 has a longest side of 7.
True or false?
Answer:
11.Which length can be drawn as the longest side of one right-angled triangle whose shorter sides are whole numbers?
- A. 22
- B. 25
- C. 11
- D. 6
Choose the right answer
Answer:
12.5 can be drawn with one right-angled triangle whose shorter sides are whole numbers.
True or false?
Answer:

13.A square-root spiral carries on from a longest side of 13, adding a side of 1 each time. Put each step's working in order.
- A. 16 + 1² = 17
- B. 15 + 1² = 16
- C. 13 + 1² = 14
- D. 14 + 1² = 15
Write the letters in the right order
Working:
Answer:
14.A rectangle is 1 cm long and 1 cm wide. Its diagonal is cm, between cm and cm (two whole numbers in a row).
Fill in each blank
Working:
Answer:
15.A square with 1 cm sides has a diagonal of 2 cm. A 2nd square is drawn on that diagonal, a 3rd on the 2nd square's diagonal, and so on. What is the area of the 9th square?
Working:
Answer:
16.To find 45 on a number line, draw a right-angled triangle with shorter sides 3 and , then an arc about 0 with the longest side as radius.
Working:
Answer:
17.Ka-yee says a right-angled triangle with shorter sides 7 cm and 3 cm draws a line 100 cm long. Ka-yee is right.
True or false?
Working:
Answer:
18.On squared paper (each square 1 cm), Tsz-ching draws 3 squares right and then 9 squares up. The straight distance from start to finish is cm.
Working:
Answer:

Show the answers and working

Constructing lengths √n worksheet: Answers and working

1. 18
The longest side squared = 3² + 3² = 18, so it is 18.
2. 64, 36, 100
8² = 64 and 6² = 36, which add to 100, so the longest side is 100.
3. 41
The longest side squared = 4² + 5² = 41, so it is 41.
4. 6
1² + □² = 37, so □² = 37 − 1² and □ = 6.
5. 11
The 1st has 2, and each new triangle adds 1 under the root: the 10th has 11.
6. 3
4² + □² = 25, so □² = 25 − 4² and □ = 3.
7. 3
(5)² + □² = 14, so 5 + □² = 14, □² = 14 − 5 and □ = 3.

8. D. 2 and 4
2² + 4² = 20, so shorter sides 2 and 4 give 20.
9. True
Square the two shorter sides before adding: 4² + 4² = 32, so 32.
10. False
Square the two shorter sides before adding: 5² + 2² = 29, so 29.
11. B. 25
25 = 4² + 3², so shorter sides 4 and 3 draw 25.
12. True
Look for positive whole numbers a and b with a² + b² = 5: try pairs from 1, 4, 9, 16, 25, 36, 49.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. C → D → B → A
Each step adds 1² to the last longest side squared: 14, 15, 16, 17.
14. 2, 1, 2
The diagonal makes two right-angled triangles: diagonal squared = 1² + 1² = 2, and 1² < 2 < 2².
15. 256 cm²
A square's diagonal squared = side² + side², twice its area, so each new square doubles the last: the 9th is 2^(9 − 1) = 256.
16. 6
3² + □² = 45, so □² = 45 − 3² and □ = 6.
17. False
The longest side squared is the sum of the two squares, 7² + 3² = 58, not (7 + 3)².
18. 90
The two moves are a right-angled triangle's shorter sides: the distance squared = 3² + 9² = 90, so 90 cm.

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