Pythagoras' theorem · Ages 13–14
Using Pythagoras' theorem
Find the right triangle and its hypotenuse, then add or subtract.
How to do it
Method
Find the right triangle and its hypotenuse, then add or subtract
Ladder 13 m, foot 5 m out: height = 132 − 52 = 12 m
Another way
Spot a hidden triple
5, ?, 13 → 5, 12, 13
In full
In a word problem draw the picture first and find the right-angled triangle: a ladder against a wall, the wall upright on level ground, the ladder the hypotenuse; a rectangle's diagonal cuts it into two right-angled triangles, the diagonal the hypotenuse; two points on squared paper, the squares across and the squares up the two legs. Then decide whether you want the hypotenuse (squares added, then the root) or a leg (squares subtracted, then the root). A 13 m ladder with its foot 5 m from the wall: the top is 132 − 52 = 144 = 12 m up. Keep every length in one unit. In a two-step problem, find the side two triangles share in one of them first, then use it in the other. Or spot a hidden Pythagorean triple: 5, ?, 13 is 5, 12, 13, so the height is 12 m without working.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A 2.6 m ladder leans on a wall, its foot 1 m out. Using Pythagoras' theorem for the height of its top, which length is the hypotenuse?
- A. The height of the top
- B. There is none
- C. The ladder
- D. The foot's distance from the wall
Choose the right answer
Show the answer and working
Answer: C. The ladder
The right angle is at the foot of the wall and the ladder faces it: the hypotenuse. Height = 2.62 − 12 = 2.4 m.
Example 2
In the right trapezium ABCD, AB = 12 cm, DC = 28 cm and BC = 20 cm. How long is the height AD, in cm?
Show the answer and working
Answer: 12 cm
Drop an upright from B to DC: a right triangle with hypotenuse BC = 20 cm and base 28 − 12 = 16 cm, so the upright is 202 − 162 = 400 − 256 = 12 cm, as long as AD.
Example 3
Ka-lok walks 24 m east from A, then 36 m north, then another 24 m east to B. How far is B from A in a straight line, in m?
Show the answer and working
Answer: 60 m
East comes to 24 + 24 = 48 m in all and north to 36 m: A to B is the hypotenuse on legs of 48 m and 36 m, 482 + 362 = 3600 = 60 m.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A 6.5 m ladder leans on a wall, its foot 2.5 m out. Using Pythagoras' theorem for the height of its top, which length is the hypotenuse?
- A. The ladder
- B. The foot's distance from the wall
- C. The height of the top
- D. There is none
Choose the right answer
Answer:
2.In the right trapezium ABCD, AB = 2 cm, DC = 47 cm and AD = 24 cm. Find BC. The answer is cm
Answer:
3.Each small square on the paper is 1 cm across. In the picture, AB = cm.
Answer:
4.Ming walks 7 m east from A, then 12 m north, then another 2 m east to B. How far is B from A in a straight line, in m?
Answer:
5.In triangle ABC, D is on BC with AD ⊥ BC, AD = 12 cm, BD = 16 cm and DC = 35 cm. What is the perimeter of triangle ABC, in cm?
Answer:
All the answers
- 1. A. The ladder
- 2. 51
- 3. 61
- 4. 15 m
- 5. 108 cm
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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