Construct a triangle ABC if it is given that BC=6cm, m/_ABC=30° and m/_BCA=100°.

The following are some basic facts and theorems of a triangle.

Angles and Sides of a Triangle

• The interior angles of a triangle are always added up to 180 degrees. The exterior angle of a triangle is equal to the sum of the two interior angles that are not adjacent to it.
   
• The sum of the lengths of any two sides of a triangle is always larger than the length of the third.
   
• Pythagorean theorem—in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. Conversely, if the lengths of the sides of a triangle satisfy the above condition, the angle opposite to the longest side is a right angle.
   
• Sine rule—the ratio of the length of a side to the sine of its opposite angle is constant
  
        x/sin(b) = y/sin(a) = z/sin(c) 
   
• The angle of a triangle can be calculated from its sides
  
        a = arccos((x² + z² - y²)/2xz)
        b = arccos((y² + z² - x²)/2yz)
        c = arccos((x² + y² - z²)/2xy)

Area of a Triangle:

• If the length of the base and the height are known.
  
        Area = ½bh
  
  Where b is the length of the base and h is the height on the base.
   
• If length of two sides and the angle between them are known.
  
        Area = ½xy sin(c) = ½xz sin(a) = ½yz sin(b) 
   
• Heron's formula—if the length of the three sides are known.
  
        Area = (s(s-x)(s-y)(s-z))^½
• where s = ½(x + y + z)

Step 1 Start by writing out the Sine Rule formula for finding sides:       a = b sin(A) sin(B) Step 2 Fill in the values you know, and the unknown length:       x = 6 sin(30°) sin(100°) Remember that each fraction in the Sine Rule formula should contain a side and its opposite angle. Step 3 Solve the resulting equation to find the unknown side, giving your answer to 3 significant figures:   x = 6      (multiply by sin(80°) on both sides) sin(30°) sin(100°)   x = 7 × sin(30°) sin(100°)   x = 3.55399314160   Note that you should try and keep full accuracy until the end of your calculation to avoid errors.