Oxford University Maths Admissions Test 2023 Question 8

University of Oxford Mathematician Dr Tom Crawford answers each Oxford Maths Admissions Test question in under 60 seconds. This is Q8 from the 2023 MAT.

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[Jaeghead]

Another approach:

By Pythagoras one altitude of those triangles has length sqrt(100 - c²/4), where c is the third length that is varying. The area is then (c/2)sqrt(100 - c²/4), but we can maximize its square instead: (c²/4)(100 - c²/4). The polynomial x(100 - x) is maximal when x = 50, so solving c²/4 = 50 gives us c = 10sqrt(2) which is closest to (d).

[adwz]

Your videos are proof that all MAT MCQs can be done under 2 minutes each, given you see the right trick properly. In this example, we see all the options concern two side lengths of size 10, so the best bet is indeed to use the sin-area formula and those two sides

[duzyolek]

And it's quite intuitive (easy to guess) that such triangle would have the largest area!

[alonbenjo]

Nice!
Another approach (without trig): use heron's formula