Taylor designs a logo using isosceles triangles joined at a central point, P - OCR - GCSE Maths - Question 25 - 2023 - Paper 3

To find the height h of the isosceles triangle, we can use the relationship between the base b and the height h. The base of the isosceles triangle is bisected into two equal parts, making each half equal to ( \frac{b}{2} ).

Since the triangles have rotational symmetry of order 60, the angle at point P for each triangle is ( \frac{360°}{60} = 6° ). In this isosceles triangle, we can use trigonometric relationships to establish a relationship involving the height h:

Using the tangent function, we have:

$\tan(3°) = \frac{h}{\frac{b}{2}}$

Substituting the value of b:

$\tan(3°) = \frac{h}{\frac{40}{2}}$

This simplifies to:

$\tan(3°) = \frac{h}{20}$

To isolate h, multiply both sides by 20:

$h = 20 \cdot \tan(3°)$

Using a calculator, we find that ( \tan(3°) \approx 0.0524 ). Thus:

$h = 20 \cdot 0.0524 \approx 1.048$

Rounding to one decimal place gives:

$h \approx 1.0 \text{ mm}$