Pythagoras' Theorem
Senior 2 · Mathematics
Pythagoras' Theorem
Lesson 3 of 4
22 min
Whenever something is vertical and something else is horizontal, a right angle appears, and with it Pythagoras' theorem: ladders against walls, masts and their support cables, paths cutting across rectangular fields, straight-line distances on maps and on graphs.
A 5 m ladder leans against a vertical wall. Its foot is 1.4 m from the wall. How high up the wall does it reach?
The ladder is the hypotenuse. h2 + 1.42 = 52, so h2 = 25 − 1.96 = 23.04 and h = √23.04 = 4.8 m.
A school field is a rectangle 80 m long and 60 m wide. Students walk diagonally across it instead of along two sides. The diagonal is √(802 + 602) = √(6400 + 3600) = √10000 = 100 m, so the short cut saves 140 − 100 = 40 m.
From a junction, a moto rider travels 9 km east and then 12 km north. How far is she from the junction in a straight line? √(81 + 144) = √225 = 15 km.
Find the distance between P(1, 2) and Q(7, 10) on a coordinate grid.
Draw the right-angled triangle with a horizontal leg and a vertical leg. The horizontal leg is 7 − 1 = 6 and the vertical leg is 10 − 2 = 8. So PQ = √(36 + 64) = √100 = 10 units.
In general, the distance between (x1, y1) and (x2, y2) is √((x2 − x1)2 + (y2 − y1)2).
Tip
Exam technique. Always draw the sketch, even if one is given; redrawing makes you find the right angle and the hypotenuse. A ladder, a cable, a diagonal or a straight-line distance is almost always the hypotenuse. Watch the units: convert everything to metres (or everything to cm) before squaring.
A 6.5 m ladder reaches 6 m up a wall. How far is its foot from the wall?
Solution: d2 = 42.25 − 36 = 6.25, so d = 2.5 m.
A phone screen is 9 cm wide and 16 cm tall. Find its diagonal to 1 decimal place.
Solution: √(81 + 256) = √337 ≈ 18.4 cm.
Find the distance between A(−2, 3) and B(4, −5).
Solution: Horizontal leg 4 − ( − 2) = 6; vertical leg −5 − 3 = −8, length 8. AB = √(36 + 64) = 10 units.
A cyclist rides 12 km north and then 5 km east. How far is he from his starting point?
Solution: √(144 + 25) = √169 = 13 km.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
