Taylor designs a logo using isosceles triangles joined at a central point, P - OCR - GCSE Maths - Question 8 - 2023 - Paper 6

To find the height, h, of the isosceles triangle when the base, b, is given, we can use the properties of isosceles triangles and their symmetry.

1. The angle of rotational symmetry about point P is 60 degrees. Therefore, each triangle will encompass an angle of $\frac{360}{60} = 6$ degrees at point P.

2. Consider half of this triangle, as the height bisects the base. Hence, each half of the base is: $\frac{b}{2} = \frac{40}{2} = 20$ mm.

3. By using trigonometry:

   • In the right triangle formed by half the base and the height, we use the tangent function: $\tan(3 \text{ degrees}) = \frac{h}{\frac{b}{2}} = \frac{h}{20}$

4. Solving for h gives us: $h = 20 \cdot \tan(3 \text{ degrees})$

5. Using a calculator, we find that: $\tan(3 \text{ degrees}) \approx 0.0524$, thus: $h \approx 20 \cdot 0.0524 \approx 1.048$ mm.

6. Rounding to one decimal place, we find: $h \approx 1.0$ mm.