Pythagoras' Theorem
Geometry · Pythagoras' Theorem
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 8 Pythagoras' Theorem
Lesson objectives
By the end of this topic, you should be able to:
- State Pythagoras' theorem.
- Solve problems using Pythagoras' theorem.
- Apply Pythagoras' theorem to solve problems in real life situations.
- Recognise Pythagorean triples and use the converse of the theorem.
Pythagoras' Theorem
Take any right-angled triangle. The square on the hypotenuse equals the sum of the squares on the other two sides.
a) The statement
Let a and b be the two shorter sides, and c the hypotenuse. Then a² + b² = c².
The hypotenuse is always opposite the right angle, and always the longest side.
Identify the hypotenuse before doing anything else. Getting it wrong turns an addition into a subtraction.
The theorem applies only to right-angled triangles.
b) Finding the hypotenuse
Square both shorter sides, add them, then take the square root.
c) Finding a shorter side
If the hypotenuse and one short side are known, subtract instead of adding.
Take c = 25 and a = 7. Then b² = 625 − 49 = 576, so b = 24.
The hypotenuse squared is always the larger, so it comes first in the subtraction.
If a shorter side comes out longer than the hypotenuse, you added when you should have subtracted.
d) Pythagorean triples
A Pythagorean triple is a set of three whole numbers satisfying the theorem.
The common ones are 3, 4, 5 and 5, 12, 13. Also 8, 15, 17 and 7, 24, 25.
Any multiple of a triple is also a triple. So 6, 8, 10 and 9, 12, 15 both work.
Recognising these saves time. If two sides are 9 and 12, the third is 15 without any calculation.
e) The converse
The theorem works backwards. If a² + b² equals c², then the triangle must be right-angled.
If a² + b² is less than c², the angle is obtuse. If greater, it is acute.
This is how a builder checks a corner. Measure 3 m and 4 m along the two walls. If the diagonal is exactly 5 m, the corner is square.
f) In three dimensions
Pythagoras applies twice for a diagonal through a solid.
A cuboid has edges a, b and c. Its space diagonal d obeys d² = a² + b² + c².
Find the diagonal of the base first, then use that with the height.
g) Where this is used
Finding the length of a ladder against a wall. Diagonals of rectangular fields. Checking that corners are square. Distance between two points on a coordinate plane.
Words to know
- Hypotenuse -- the side opposite the right angle, always the longest side.
- Pythagorean triple -- three whole numbers satisfying a² + b² = c².
- Converse -- the reverse statement, used to test for a right angle.
- Space diagonal -- the longest diagonal of a cuboid, corner to opposite corner.
- Leg -- either of the two shorter sides of a right-angled triangle.
:::checkpoint Check yourself
- A triangle has short sides 8 cm and 15 cm. Find the hypotenuse.
- A triangle has hypotenuse 26 cm and one short side 10 cm. Find the other.
- Do sides 6, 8 and 11 form a right-angled triangle?
- Why is the hypotenuse always the longest side? :::
Bridge to practice
The exercises begin with the statement and finding the hypotenuse, move through finding shorter sides, triples and the converse, and finish with real applications and three-dimensional problems. Before every calculation, draw the triangle and mark which side is the hypotenuse.