Use surds for exact lengths, areas and distances, and for quadratic solutions in the form a ± b√c.
What you will learn
- Give exact lengths by Pythagoras
- Work out areas and perimeters with surd lengths
- Solve quadratics in surd form
- Solve equations with surd coefficients
"Give your answer as a surd"
Many exact lengths come from Pythagoras' theorem.
In a right-angled triangle the longest side, opposite the right angle, is the hypotenuse.
If the hypotenuse is c and the other two sides are a and b, then c² = a² + b². For a shorter side, subtract: a² = c² − b².
Do not round. In a right-angled triangle with shorter sides 4 cm and 6 cm, the hypotenuse is √(16 + 36) = √52 = 2√13 cm, exactly.
The distance between (1, 2) and (5, 8) is √(4² + 6²) = √52 = 2√13 in the same way: 4 is the change in x (5 − 1) and 6 the change in y (8 − 2).
A shorter side: hypotenuse 8 cm and one side 4 cm, so the other side is √(64 − 16) = √48 = 4√3 cm.
The diagonal of a square of side 5√2 cm is √(50 + 50) = √100 = 10 cm. A rectangle √3 cm by √5 cm has diagonal √(3 + 5) = √8 = 2√2 cm.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- "Give your answer as a surd"
- Areas and perimeters with surds
- Quadratics in the form a ± b√c
- Try it: Exact and decimal side by side
7 practice questions, marked as you go; about 25 minutes in all.
Key terms
Pythagoras' theorem, Hypotenuse
In this unit: Surds
Exact answers with square roots: simplifying, multiplying and expanding, rationalising denominators, and surds in Pythagoras and quadratics. Higher tier.
- Simplifying surds
- Multiplying and expanding surds
- Rationalising the denominator
- Exact answers with surds