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

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Taylor designs a logo using isosceles triangles joined at a central point, P. This is the start of Taylor's design. The completed design will have rotational symme... show full transcript

Worked Solution & Example Answer:Taylor designs a logo using isosceles triangles joined at a central point, P - OCR - GCSE Maths - Question 8 - 2023 - Paper 6

Step 1

Calculate h when b = 40 mm

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Answer

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 36060=6\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: b2=402=20\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 degrees)=hb2=h20\tan(3 \text{ degrees}) = \frac{h}{\frac{b}{2}} = \frac{h}{20}
  4. Solving for h gives us: h=20tan(3 degrees)h = 20 \cdot \tan(3 \text{ degrees})

  5. Using a calculator, we find that: tan(3 degrees)0.0524\tan(3 \text{ degrees}) \approx 0.0524, thus: h200.05241.048h \approx 20 \cdot 0.0524 \approx 1.048 mm.

  6. Rounding to one decimal place, we find: h1.0h \approx 1.0 mm.

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