Make an accurate drawing of the triangle and measure length d to 1 d.p.

$$(7.5)^2 = 56.25$$ $$(8.9)^2 = 79.21$$

$$d^2 = 56.25 + 79.21 - 133.5 \times 0.20791$$ $$d^2 = 135.46 - 27.750285$$ $$d^2 = 107.709715$$

$$d = \sqrt{107.709715}$$ $$d \approx 10.3783$$

$$d \approx 10.4 \text{ cm}$$

To make an accurate drawing of the triangle:

1. Draw a line segment of length 8.9 cm.
2. At one end of this line segment, use a protractor to measure and mark an angle of $78^\circ$.
3. From the vertex of the $78^\circ$ angle, draw a line segment of length 7.5 cm along the $78^\circ$ mark.
4. Connect the end of the 8.9 cm line segment to the end of the 7.5 cm line segment. This third side is $d$.
5. Measure the length of side $d$ with a ruler. It should be approximately 10.4 cm.

The measured length $d$ is:

$$d = \boxed{\text{10.4 cm}}$$