a. Pythagoras theorem
AD and AB are known.
Solve for BD.
b. Tan(∠ABD) = opposite/adjacent = DA/AB
Solve for ∠ABD.
Since ∠ABD and ∠DBC form a straight line, they are supplementary (add up to 180°).
Solve for ∠DBC:
∠DBC = 180 - ∠ABD
c. Sine rule
The sine rule applies when all angles are known plus one side, or two sides are known and one angle which is not the included angle.
The given triangle applies in to the first case.
BD, ∠DBC, ∠BCD are known, so apply the sine rule:
where A is the angle opposite side a, etc.
Solve for DC.
d. The cosine rule applies when two adjacent sides and the included angle are known.
By now, BD, DC and ∠BDC are known.
Note: ∠BDC = 180-(∠DBC+∠BCD)
So apply the cosine rule to solve for c
BC²=BD²+CD²-2 BD CD cos(∠BDC)
Post to check your results if you wish.
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