15 (a) Calculate length DF in this triangle - OCR - GCSE Maths - Question 15 - 2019 - Paper 4

To find the length DF, we can use the cosine rule. In triangle DFG:

1. Apply the cosine rule: $DF^2 = DG^2 + FG^2 - 2 \cdot DG \cdot FG \cdot \cos(30^\circ)$ where,

   • $DG = 12 \, cm$
   • $FG = 10 \, cm$

2. Substitute the values into the formula: $DF^2 = 12^2 + 10^2 - 2 \cdot 12 \cdot 10 \cdot \cos(30^\circ)$

3. Calculate:

   • First, find ( \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 ).
   • Substitute: $DF^2 = 144 + 100 - 240 \cdot 0.866$ $DF^2 = 244 - 207.84 \\$ $DF^2 = 36.16 \\$ $DF = \sqrt{36.16} \approx 6.01 \, cm$

Thus, the length DF is approximately $6.01 \, cm$.