Answer:
To find the length of CD, we can use the cosine rule in triangle BCD:
[tex]\[CD^2 = BC^2 + BD^2 - 2 \times BC \times BD \times \cos(\angle BCD)\][/tex]
Given that BC = 16 cm, BD = 12 cm, and [tex]\angle BCD = 40°[/tex], we can plug these values into the formula:
[tex]\[CD^2 = 16^2 + 12^2 - 2 \times 16 \times 12 \times \cos(40°)\][/tex]
Now, let's calculate:
[tex]\[CD^2 = 256 + 144 - 384 \times \cos(40°)\][/tex]
[tex]\[CD^2 = 400 - 384 \times \cos(40°)\][/tex]
[tex]\[CD^2 = 400 - 384 \times 0.766\] (cos(40°) ≈ 0.766)[/tex]
[tex]\[CD^2 ≈ 400 - 294.144\][/tex]
[tex]\[CD^2 ≈ 105.856\][/tex]
[tex]\[CD ≈ \sqrt{105.856}\][/tex]
[tex]\[CD ≈ 10.288\][/tex]
Rounded to three significant figures, the length of CD is approximately 10.3 cm.
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