# Puzzles

## Largest triangle

What is the largest area triangle which has one side of length 4cm and one of length 5cm?

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As our shape is a triangle, the 4cm and 5cm sides must be adjacent. Call the angle between them be \(\theta\).

The area of the triangle is \(\frac{1}{2}\times 4\times 5 \times \sin{\theta}\) or \(10\sin{\theta}\). This has a maximum value when \(\theta=90^\circ\), so the largest triangle has and area of 10cm^{2} and looks like:

#### Extension

What is the largest area triangle with a perimeter of 12cm?