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

To determine the height, h, of the isosceles triangle, we can use the properties of the triangle in relation to its base.

1. Base and Height Relationship: For isosceles triangles, we can split the triangle down the middle. This creates two right triangles, where:

   • Each triangle's base will be half of b, that is, rac{b}{2}.
   • The height of the triangle is represented by h.

   So we have: $h^2 + \left( \frac{b}{2} \right)^2 = \text{(hypotenuse)}^2$

   Given that the logo has rotational symmetry of order 60, the angle at point P for each triangle is: $\theta = \frac{360}{60} = 6 \text{ degrees}$

2. Finding h: For each of the two right triangles formed, using trigonometric ratios, we can state: $\tan(\theta) = \frac{h}{\frac{b}{2}}$ Rearranging gives: $h = \tan(6^{\circ}) \times \frac{b}{2}$

   Substituting $b = 40 mm$: $h = \tan(6^{\circ}) \times \frac{40}{2} = \tan(6^{\circ}) \times 20$

   Calculate:

   • Using a calculator, find $\tan(6^{\circ})$ which is approximately 0.1051.
   • Therefore: $h \approx 0.1051 \times 20 \approx 2.102$

3. Final Answer: Rounding this answer to 1 decimal place gives: $h \approx 2.1 \text{ mm}$