The diagram shows a triangle, ABC, with perpendicular height BD - OCR - GCSE Maths - Question 14 - 2023 - Paper 5

To find the length of BD, we can use trigonometric ratios. We know the angle BAD is 45° and BC is the adjacent side to angle BAD.

Using the tangent function, we have:

$\tan(45°) = \frac{BD}{AB}$

Since (\tan(45°) = 1), this simplifies to:

$BD = AB$

Next, we can find AB using the triangle properties. We also know angle BCD is 30°. In triangle BCD:

$\tan(30°) = \frac{BD}{12}$

From the tangent function:

$BD = 12 \cdot \tan(30°) = 12 \cdot \frac{1}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3}$

Thus, the length of BD is:

$BD = 4\sqrt{3} \text{ cm}$