Geometry and proof · Ages 13–14

Using Pythagoras' theorem worksheet

Part of: Pythagoras' theorem

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Geometry and proof · Ages 13–14

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Name:Date:Score:／48

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

1.A 3 m ladder leans against a wall with its foot 1.8 m from the wall. How high up the wall is its top, in m?
Answer:
2.Rectangle ABCD has sides of 16 cm and 12 cm. How long is its diagonal AC, in cm?
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3.Rectangle ABCD has sides of 6 cm and 8 cm. How long is its diagonal AC, in cm?
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4.Each small square on the paper is 1 cm across. In the picture, AB = cm.
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5.A rectangular lawn is 24 m by 18 m. Walking along two edges from one corner to the opposite one is 42 m. How many m shorter is it straight along the diagonal?
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6.On the map each small square stands for 20 m. How far is it in a straight line from A to B, in m?
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7.In the right trapezium ABCD, AB = 8 cm, DC = 35 cm and AD = 120 cm. Find BC. The answer is cm
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8.A rectangular lawn is 30 m by 72 m. Walking along two edges from one corner to the opposite one is 102 m. How many m shorter is it straight along the diagonal?
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9.In triangle ABC, D is on BC with AD ⊥ BC, AD = 12 cm, BD = 16 cm and DC = 5 cm. What is the perimeter of triangle ABC, in cm?
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10.In the right trapezium ABCD, AB = 7 cm, DC = 16 cm and BC = 15 cm. How long is the height AD, in cm?
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11.Ka-lok walks 72 m east from A, then 48 m north, then another 18 m east to B. How far is B from A in a straight line, in m?
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13.A pitched roof's cross-section is an isosceles triangle 28 m wide at the base, each sloping side 50 m. The roof is m high and the cross-section is m².
Fill in each blank
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14.An old television's screen has width and height in the ratio 4 : 3 and a diagonal of 35 cm. The screen is cm wide and cm high.
Fill in each blank
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15.Two boats leave the pier together, one sailing east at 25 km an hour, the other north at 60 km an hour. How far apart are they after 4 hours, in km?
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16.On the map each small square stands for 20 m. From A to B is 4 squares east and 1 squares north. About how far apart are A and B in a straight line, in m, to 1 decimal place?
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17.A box is 2 cm long, 8 cm wide and 16 cm high. The square of its base's diagonal is cm², and the straight distance from a bottom corner to the farthest top corner is cm.
Fill in each blank
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18.A rectangular lawn has a diagonal of 10 m and one side of 6 m. How many m of railing go all the way round it?
Working:
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Show the answers and working

Using Pythagoras' theorem worksheet: Answers and working

1. 2.4 m
The wall meets the ground at a right angle and the ladder is the hypotenuse: 3² − 1.8² = 9 − 3.24 = 5.76, so the height = 5.76 = 2.4 m.
2. 20 cm
The diagonal cuts the rectangle into two right-angled triangles and is their hypotenuse: 16² + 12² = 400, so AC = 400 = 20 cm.
3. 10 cm
The diagonal cuts the rectangle into two right-angled triangles and is their hypotenuse: 6² + 8² = 100, so AC = 100 = 10 cm.
4. 17
AB is the hypotenuse of a right triangle whose legs run 1 squares across and 4 up: 1² + 4² = 1 + 16 = 17, so AB = 17 cm.
5. 12 m
The diagonal = 24² + 18² = 900 = 30 m, so it saves 42 − 30 = 12 m.
6. 100 m
4 squares across and 3 up, 80 m and 60 m: 80² + 60² = 100 m (20 times 3, 4, 5).
7. 123
Drop an upright from B to DC: a right triangle with legs 35 − 8 = 27 cm and the height 120 cm, so BC = 27² + 120² = 123 cm.
8. 24 m
The diagonal = 30² + 72² = 6084 = 78 m, so it saves 102 − 78 = 24 m.
9. 54 cm
AB = 12² + 16² = 20 cm and AC = 12² + 5² = 13 cm; the perimeter = 20 + 13 + 16 + 5 = 54 cm.
10. 12 cm
Drop an upright from B to DC: a right triangle with hypotenuse BC = 15 cm and base 16 − 7 = 9 cm, so the upright is 15² − 9² = 225 − 81 = 12 cm, as long as AD.
11. 102 m
East comes to 72 + 18 = 90 m in all and north to 48 m: A to B is the hypotenuse on legs of 90 m and 48 m, 90² + 48² = 10404 = 102 m.

12. B. The ladder
The right angle is at the foot of the wall and the ladder faces it: the hypotenuse. Height = 3.5² − 2.1² = 2.8 m.

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

13. 48, 672
The height drops from the top to the middle of the base, halving it to 14 m: height = 50² − 14² = 48 m; area = 28 × 48 ÷ 2 = 672 m².
14. 28, 21
Say the width is 4k and the height 3k: (4k)² + (3k)² = (5k)², so the diagonal 5k = 35 and k = 7; 28 cm wide, 21 cm high.
15. 260 km
After 4 hours one is 100 km east and the other 240 km north, at right angles: the distance = 100² + 240² = 260 km.
16. 82.5 m
80 m east and 20 m north: the distance = 80² + 20² = 6800 ≈ 82.5 m.
17. 68, 18
The base's diagonal² = 2² + 8² = 68; it meets the height at a right angle, so the distance = 68 + 16² = 18 cm.
18. 28 m
The other side = 10² − 6² = 8 m; the perimeter = 2 × (6 + 8) = 28 m.

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