2020HSCShort answer

Q13 (c) (i)

2 marks

Question

By considering the right-angled triangle below, or otherwise, prove that
$\frac{a+b}{2}\geq\sqrt{ab},$
where $a>b\geq0$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Applies Pythagoras to find $x$ , or equivalent merit<br>OR<br>Observes that $x\le a+b$	1

Sample answer:

By Pythagoras,

(a+b)^2=x^2+(a-b)^2

\therefore x=2\sqrt{ab}

Now the hypotenuse is the longest side, so

a+b\ge2\sqrt{ab}

\therefore\frac{a+b}{2}\ge\sqrt{ab}

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Compare this with your working, not only your final sentence.

Syllabus alignment

Working MathematicallyProof

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.

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